Content-Type: multipart/mixed; boundary="-------------0907121236356"

This is a multi-part message in MIME format.

---------------0907121236356
Content-Type: text/plain; name="09-114.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="09-114.keywords"

Mather's theory, beta function, integrability
---------------0907121236356
Content-Type: application/x-tex; name="Article_Diffbeta#FinalVersion10_0_09fileps#.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="Article_Diffbeta#FinalVersion10_0_09fileps#.tex"

\documentclass[10pt]{article}
\setlength{\oddsidemargin}{0cm}
\setlength{\evensidemargin}{0cm}
\setlength{\textwidth}{14.5cm}

\usepackage{amsmath,amssymb,amsbsy,amsfonts,latexsym,amsthm,amscd,amsxtra,
amssymb}
\usepackage[all]{xy}
%\usepackage{pdftricks}
\usepackage{graphicx}
\usepackage{epstopdf}



%\parindent 0pt

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Definitions, Theorems etc...%%%%%%%%%%%%%%%%%%
\newtheorem{sez}{}[section]
\newtheorem{Teo}{Theorem}%[section]
\newtheorem{Def}{Definition}
\newtheorem{Prop}{Proposition}
\newtheorem{Cor}{Corollary}
\newtheorem{Lem}{Lemma}
\newtheorem{Rema}{Remark}
\newenvironment{Rem}{\begin{Rema} \begin{upshape}} {\end{upshape}\end{Rema}}
\newtheorem*{Pf}{Proof}
\newenvironment{Proof}{\begin{Pf} \begin{upshape}} {\end{upshape} \qed\end{Pf}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Equazioni %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand\beqa[1]{ \begin{eqnarray} \label{#1}}
\newcommand{\eeqa}{ \end{eqnarray} }
\newcommand{\beqano}{ \begin{eqnarray*} }
\newcommand{\eeqano}{ \end{eqnarray*} }


\newcommand{\T}{ {\mathbb T}   }
\newcommand{\N}{ {\mathbb N}   }
\newcommand{\R}{ {\mathbb R}   }
\newcommand{\Z}{ {\mathbb Z}   }
\newcommand{\C}{ {\mathbb C}   }
\newcommand{\Q}{ {\mathbb Q}   }
\newcommand{\tS}{ {\mathbb S}   }
\newcommand{\K}{ {\mathbb K}   }
%\renewcommand{\L}{ {\mathbb L}   }

\font\msytw=msbm9 scaled\magstep1
\font\cs=cmcsc10

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Greek letters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\renewcommand \a {\alpha}
\newcommand \e {\varepsilon }
\newcommand \x {\xi}
\renewcommand \b  {\beta}
\renewcommand \r {\rho}
\renewcommand \d {\delta}
\newcommand \D{\Delta}
\newcommand \m {\mu}
\newcommand \tm {\tilde{\m}}
\newcommand \n {\nu}
\newcommand \om {\omega}
\newcommand \Om {\Omega}
\newcommand \ff {\phi}
\newcommand \f {\varphi}
\newcommand \F {\Phi}
\newcommand \g {\gamma}
\newcommand \tg {\tilde{\gamma}}
\newcommand \G {\Gamma}
\newcommand \s {\sigma}
%\renewcommand \S {\Sigma}
\renewcommand \k {\chi}
\renewcommand \t {\tau}
\renewcommand \th {\theta}
\renewcommand \l {\lambda}
\renewcommand \L {\Lambda}
\newcommand \ph {\phi}
\newcommand \Ph {\phi}
\newcommand \et {\eta}
\newcommand \z {\zeta}
\newcommand \p {\pi}
\renewcommand \P {\Pi}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Various %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\renewcommand \c[1]{\mathcal{#1}}
\newcommand \cA {{\mathcal A}}
\newcommand \cC {{\mathcal C}}
\newcommand \cE {{\mathcal E}}
\newcommand \cF {{\mathcal F}}
\newcommand \cL {{\mathcal L}}
\newcommand \cM {{\mathcal M}}
\newcommand \cN {{\mathcal N}}
\newcommand \cR {{\mathcal R}}
\newcommand \cS {{\mathcal S}}
\newcommand \cT {{\mathcal T}}

\newcommand \calM {\mathfrak{M}}
\newcommand \fg {{\mathfrak g}}
\def\ie{\hbox{\it i.e.,\ }}
\def\eg{\hbox{\it e.g.,\ }}



\newcommand \dpr {\partial}
\newcommand \ov {\overline}
\newcommand \rH {{\rm H}}
\newcommand \rT {{\rm T}}
\newcommand \supp {{\rm supp\ }}
\newcommand \rank {{\rm rank\ }}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{Differentiability of Mather's average action and integrability on closed surfaces}
\author{\sc{Daniel Massart and Alfonso Sorrentino}}
\date{}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%
\begin{center}
\parbox[c]{4.5 in}
{\small In this article we study the differentiability of Mather's $\beta$-function on closed surfaces and its relation to the integrability of the system.}\\
\end{center}

\begin{center}
\parbox[c]{4.5 in}
{\small Dans cet article nous \'etudions la diff\'erentiabilit\'e de la fonction $\beta$ de Mather sur surfaces 
et ses cons\'equences sur l'int\'egrabilit\'e du syst\`eme.}\\
\end{center}


\vspace{5pt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}\label{section1}

In the study of Tonelli Lagrangian and Hamiltonian systems,  a central role in understanding  the dynamical and topological properties of the  action-minimizing sets (also called {\it Mather} and {\it Aubry sets}), is played by the so-called {\it Mather's average action} (sometimes referred to as $\beta$-{\it function}  or {\it effective Lagrangian}), with particular attention to its differentiability and non-differentiability properties. %and its convex conjugate $\alpha$ function.
Roughly speaking,  this is a convex superlinear function on the first homology group of the base manifold, which represents the minimal action of invariant probability measures within a prescribed  {\it homology class}, or {\it rotation vector} (see Equation (\ref{defbeta}) for a more precise definition). 
Understanding whether or not this function is differentiable, or even smoother, %what happens at non-differentiability points, or 
and what are the implications of its regularity to the dynamics  of the system is a formidable problem, which is still far from being completely understood.
Examples of Lagrangians  admitting a smooth $\beta$-function are easy to construct. Trivially, if the base manifold $M$ is such that $\dim \rH_1(M;\R)=0$ then $\beta$ is a function defined on a single point and it is therefore smooth. Furthermore, if $\dim \rH_1(M;\R)=1$ then a result by M. Dias Carneiro \cite{Carneiro}  allows one to conclude that $\beta$ is differentiable everywhere, except possibly at the origin. 
As soon as  $\dim \rH_1(M;\R)\geq 2$ the situation becomes definitely less clear and the smoothness of $\beta$ becomes a more ``untypical'' phenomenon. Nevertheless, it is still possible to find some interesting examples in which it is smooth.
 For instance, let 
$H:\rT^*\T^n \longrightarrow \R$ be a completely integrable (in the sense of Liouville) Tonelli Hamiltonian system, given by $H(x,p)=h(p)$, and consider the associated Lagrangian $L(x,v)=\ell(v)$ on $\rT \T^n$.  It is easy to check that in this case, up to identifying $\rH_1(\T^n;\R)$ with $\R^n$, one has $\beta(h)=\ell(h)$ and therefore $\b$ is as smooth as the Lagrangian and the Hamiltonian are.  One can weaken the assumption on the completely integrability of the system and consider $C^0$-{\it integrable systems}, \ie Hamiltonian systems that admit a foliation of the phase space by disjoint invariant continuous Lagrangian graphs, one for each possible cohomology class (see Definition \ref{C0integrability}). It is then possible to prove that in this case the associated $\beta$ function is $C^1$ (see Lemma \ref{C_zero_int_beta_C1}). 
These observations arise the following question: with the exception of the mentioned trivial cases, does the regularity of $\b$ imply the integrability of the system?




In this article we address the above problem in the case of Tonelli Lagrangians on closed surfaces, not necessarily orientable. We prove the following.\\

\noindent{\bf Main Results}. {\it
Let $M$ be a closed surface and $L:\rT M \longrightarrow \R$ a Tonelli Lagrangian. 
\begin{itemize}
\item[{\rm (i)}]   If $M$ is not the sphere, the projective plane, the Klein bottle or the torus, then $\beta$ cannot be $C^1$ on $H_1 (M,\R)$ {\rm [Proposition \ref{maintheo2}].}
\item[{\rm (ii)}] If $M$ is  the sphere, the projective plane or the Klein bottle, then the Lagrangian cannot be $C^0$-integrable {\rm [Proposition \ref{torus_only_integrable}].}
\item[{\rm (iii)}]	If $M$ is the torus, then $\beta$ is $C^1$ if and only if the system is $C^0$-integrable {\rm [Theorem \ref{maintheo1}].}\\
\end{itemize}
}

\noindent Moreover in Corollary \ref{corollarymaintheo} we shall deduce several information on the dynamics of $C^0$-integrable systems and in section \ref{mechanicalcase} we discuss the case of mechanical Lagrangians on the two-torus and  show that  in this case: $\beta$ {\it is $C^1$ if and only if the  system is the geodesic flow associated to a flat metric on $\T^2$} [Proposition \ref{prop_mechanical_2dim}].


\vspace{15 pt}

\noindent{\bf Acknowledgements.} 
The authors are particularly grateful to Victor Bangert, Patrick Bernard,  and Albert Fathi for several useful discussions.
This work has been partially funded by the ANR project  ``{\it Hamilton-Jacobi et th\'eorie KAM faible}''. A.S. also wishes to acknowledge the support of {\it Fondation des Sciences Math\'ematiques de Paris}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Mather's average action and Aubry-Mather theory}


Let us start by recalling some basic facts about Mather's theory for Tonelli Lagrangians.
Let $M$ be a compact and connected smooth $n$-dimensional manifold without boundary. Denote by ${\rm T}M$ its tangent bundle and ${\rm T}^*M$ the cotangent one and denote points in ${\rT M}$ and ${\rT^*M}$
respectively by $(x,v)$ and $(x,p)$. We shall also assume that the cotangent bundle $\rT^*M$ is equipped with the canonical symplectic structure, which we shall denote $\omega$.
A {\it Tonelli Lagrangian} is a $C^2$ function $L: \rT M \rightarrow \R$, which is strictly convex and uniformly superlinear in the fibers; in particular this Lagrangian defines a flow on ${\rT M}$, known as {\it Euler-Lagrange flow} $\Phi^L_t$, whose integral curves are solutions of $\frac{d}{dt}\frac{\dpr L}{\dpr v}(x,v)= \frac{\dpr L}{\dpr x}(x,v)$. 
Let $\calM(L)$ be the space of compactly supported probability measures on 
${\rm T}M$  invariant  under the Euler-Lagrange flow of $L$. To every 
$\m \in \calM(L)$, we may associate its {\it average action}
$$ A_L(\m) = \int_{{\rm T}M} L\,d\m\,. $$
It is quite easy to check that since 
$\m$ is invariant under 
the Euler-Lagrange flow, then for each $f\in C^1(M)$ we have 
$\int{{df}(x)\cdot v }\, d\m =0$. Therefore we can define a linear functional
\beqano
{\rm H}^1(M;\R) &\longrightarrow& \R \\
c &\longmapsto& \int_{{\rm T}M} {\eta_c}(x)\cdot v\, d\m\,,
\eeqano
where $\eta_c$ is any closed $1$-form on $M$ whose cohomology class is $c$. By 
duality, there 
exists $\rho (\m) \in {\rm H}_1(M;\R)$ such that
$$
\int_{{\rm T}M} {\eta_c}(x)\cdot v\, d\m = \langle c,\rho(\m) \rangle
\qquad \forall\,c\in {\rm H}^1(M;\R)$$ 
(the bracket on the right--hand side denotes the canonical pairing between 
cohomology and 
homology). We call $\rho(\m)$ the {\it rotation vector} of $\m$. 
%One can show that the action functional $A_L: \calM(L)  \longrightarrow \R\cup \{+\infty\}$ is lower semicontinuous and that for every $h\in {\rm H}_1(M;\R)$ there exists $\m\in \calM(L)$ with $\rho(\m)=h$. In other words
It is possible to show \cite{Mather91} that the map $\rho: \calM(L) \longrightarrow {\rm H}_1(M;\R)$ is surjective and hence there exist invariant probability measures with any given rotation vector.
Let us consider the minimal value of the average action $A_L$ over the set of 
probability measures with rotation vector $h$ (this minimum exists because of the lower semicontinuity of the action functional):
\begin{eqnarray}
\b: {\rm H}_1(M;\R) &\longrightarrow& \R \nonumber\\
h &\longmapsto& \min_{\m\in\calM(L):\,\rho(\m)=h} A_L(\m)\,. \label{defbeta}
\end{eqnarray}
%In fact, the minimum above is actually achieved (see \cite{Mather91}).
This function $\beta$ is what is generally known as {\it Mather's 
$\beta$-function}.
A measure $\m \in \calM(L)$ realizing such a minimum amongst all invariant probability measures with the same rotation vector, \ie $A_L(\m) = 
\b(\rho(\m))$, is called an {\it action minimizing} {\it measure with rotation vector} $\rho(\mu)$. 
The $\beta$-function is convex, and therefore one can consider its {\it conjugate} 
function (given by Fenchel's duality)
$ \a: {\rm H}^1(M;\R) \longrightarrow \R $ defined by
\beqano
\a(c) &:=& \max_{h\in {\rm H}_1(M;\R)} \left(\langle c,h \rangle - \b(h) 
\right)
= - \min_{h\in {\rm H}_1(M;\R)} \left(\b(h) - \langle c,h \rangle\right) =\\
&=& - \min_{\m \in \calM(L)} \left( A_L(\m) - \langle c, \rho(\m)\rangle\right)
= - \min_{\m \in \calM(L)} A_{L-\eta_c}(\m),\\
\eeqano
where $\eta_c$ is any smooth closed $1$-form on $M$ with cohomology class $c$. Observe that  the modified Lagrangian $L-\eta_c$  ($\eta_c$ can be also seen as  a function on $\rT M$: $\eta_c(x,v):=\eta_c(x)\cdot v$)  is still of Tonelli type and has the same Euler-Lagrange flow as $L$ (it follows easily from the closedness of $\eta_c$). Nevertheless, invariant probability measures that minimize the action functional  $A_{L-\eta_c}$ change according to the chosen cohomology class.
A measure  $\m\in\calM(L)$ realizing the minimum of  $A_{L-\eta_c}$ (withouth any constraint on the rotation vector) is called {\it $c$-action minimizing}.
This leads to the definition of a first important family of invariant subsets of $\rT M$: 
%{\it Mather sets}. 

\begin{itemize}
\item for a homology class $h\in \rH_1(M;\R)$, we define  the {\it Mather set of 
rotation vector} $h$ as:
%
$$ \widetilde{\cM}^h := \bigcup \left\{ {\rm supp}\,\m: \; \m\; \mbox{is action minimizing with rotation vector }\; h\right\};
$$

\item for a cohomology class $c \in {\rm H}^1(M;\R)$, we define the 
{\it Mather set of 
cohomology class} $c$ as:
%
$$ \widetilde{\cM}_c := \bigcup \left \{ {\rm supp}\,\m: \; \m\; {\rm is }\;c-\mbox{action minimizing}\right\}.
$$
\end{itemize}
 
 
 
 
The relation between these sets is described in the following lemma. To state it, let us recall that, like any convex 
function on a finite-dimensional space, Mather's   $\b$ function 
admits a subderivative at any point $h\in \rH_1(M;\R)$, \ie we can find $c\in 
\rH^1(M;\R)$ such that
$$\forall h'\in \rH_1(M;\R), \quad \b(h')-\b(h)\geq \langle c,h'-h\rangle.$$
As it is usually done, we  denote by $\partial \b(h)$ the set of $c\in 
\rH^1(M;\R)$ which are subderivatives of $\b$ at $h$, that is, the set of $c$'s which satisfy the  
above inequality (this set is also called the {\it Legendre transform of} $h$). 
By Fenchel's duality,  we have
\beqa{fenchelequality}
c\in \partial \b(h) \Longleftrightarrow \langle c,h\rangle=\a(c)+\b(h).
\eeqa


\begin{Lem}\label{lemma1}
Let $h,c$ be respectively an arbitrary homology class in $\rH_1(M;\R)$ and an arbitrary cohomology class in $ \rH^1(M;\R)$. We have
$$
{\rm (1)}\; \widetilde{\cM}^{h} \cap  \widetilde{\cM}_c \neq \emptyset 
\quad \Longleftrightarrow \quad
{\rm (2)}\; \widetilde{\cM}^{h} \subseteq \widetilde{\cM}_c \quad 
\Longleftrightarrow \quad {\rm (3)}\;c \in \partial \b(h)\,.
$$
\end{Lem}


\begin{Proof}
 The implication ${\rm (2) }\Longrightarrow {\rm (1)}$ is trivial. Let us prove that
 ${\rm  (1)} \Longrightarrow {\rm (3)}$.
If $\widetilde{\cM}^{h} \cap  \widetilde{\cM}_c\neq \emptyset $, then there exists a 
$c$-minimizing invariant measure $\m$ with rotation vector $h$. 
Let $\eta_c$ be a closed $1$-form with $[\eta_c]=c$. 
From the definitions of $\a$, $\b$ and rotation vector:
\beqano
-\a(c) = \int_{{\rm T}M} (L-{\eta_c})\,d\m = \int_{{\rm T}M} L\,d\m - 
\langle c,h\rangle = \b(h) - \langle c,h\rangle\,;
\eeqano
since $\b$ and $\a$ are convex conjugates,  then $c$ is a subderivative of 
$\b$ at $h$.\\
Finally, in order to show $(3) \Longrightarrow (2)$, let us prove that any 
action minimizing measure with rotation vector $h$ is $c$-minimizing. 
In fact, if $c \in \partial \b(h)$ then $\b(h)= \langle c,h\rangle - 
\a(c)$; therefore for any action minimizing measure $\m$ with $\rho(\m)=h$ and for any $\eta_c$ as above:
$$
-\a(c) = \b(h) - \langle c,h\rangle =  \int_{{\rm T}M}(L-{\eta_c})\, 
d\m.  
$$
This proves that $\m$ is $c$-action minimizing and it concludes the proof.
\end{Proof}


\begin{Rem}\label{mathersetsdifferentcohomology}
One can also find a relation between the Mather sets corresponding to different cohomology classes, in terms of the strictly convexity of the $\a$-function or, better, the lack thereof.
The regions where $\a$ is not strictly convex are called {\it flats}: for instance the Legendre transform of $\a$ at $c$, denoted $\dpr \a(c)$, which is the set of  homology classes for which equality (\ref{fenchelequality}) holds, is an example of flat. It is possible to check that if two cohomology classes are in the relative interior of the same flat $F$ of $\a$, then their Mather sets coincide (see \cite{Mather91, Massart03}). We denote by $\widetilde{\cM}(F)$ the common Mather set to all the cohomologies in the relative  interior of $F$. 
\end{Rem}
 
 

In addition to the Mather sets, one can also construct another interesting family of compact invariant sets. We define the {\it  Aubry set $\widetilde{\cA}_c$ of cohomology class $c$} by looking at a special kind of {\it global minimizers}: we say $(x,v) \in TM$ lies in $\widetilde{\cA}_c$ if there exists a sequence of $C^1$ curves $\gamma_n : \left[0,T_n\right] \longrightarrow M$, such that \begin{itemize}
  \item $\gamma_n (0)= \gamma_n (T_n) = x$ for all $n$
  \item $\dot{\g}_n(0)$ and $\dot{\g}_n(T_n)$ tend to $v$ as $n$ tends to infinity
       \item $T_n$ goes to infinity with $n$
  \item $\int_{0}^{T_n}  (L-\eta_c + \a(c))(\g_n(t),\dot{\g}_n(t))\,dt $ tends to $0$ as $n$ tends to infinity.
  \end{itemize}
One can prove \cite{Mather93, Fathibook} that if $(x,v)\in \widetilde{\cA}_c$, then the curve $\g(t):=\pi \circ \Phi^L_t$, where $\pi: \rT M \longrightarrow M$ denotes the canonical projection, is a $c$-{\it global minimizer}, that is, it
minimizes on any compact time interval the action of $L-\eta_c$,  among all curves with the same endpoints and time length (see \cite{Mather91}); observe, however, that not all $c$-global minimizers can be obtained in this way.




% $$ 
 %_c:= \bigcup \left\{ (\g(t),\dot{\g}(t)):\; \g\;\mbox{is a regular}\;c-%\mbox{global minimizers and}\;t\in\R \right\};
%$$
%see \cite{Mather93} for a precise definition of regular $c$-global minimizer. 
%
\noindent These action-minimizing sets that we have just defined, are such that $\widetilde{\cM}_c \subseteq \widetilde{\cA}_c$ for all $c\in\rH^1(M;\R)$ and moreover one of their most important features is that they are graphs over $M$ ({\it Mather's graph theorem} \cite{Mather91, Mather93}), \ie 
the projection along the fibers $\pi|{\widetilde{\cA}_c(L)}$ is injective and its inverse $\big(\pi|\widetilde{\cA}_c(L)\big)^{-1}\!\!\!: \pi\big(\widetilde{\cA}_c(L)\big) \longrightarrow \widetilde{\cA}_c(L)$ is Lipschitz.
%; namely, if $\pi: {\rm T}M \rightarrow M$ denotes the canonical projection along the fibers, then $\pi|{\widetilde{\cA}_c(L)}$ is injective and its inverse $\big(\pi|\widetilde{\cA}_c(L)\big)^{-1}\!\!\!: \pi\big(\widetilde{\cA}_c(L)\big) \longrightarrow \widetilde{\cA}_c(L)$ is Lipschitz. 
Hereafter we shall denote by $\cM^h$, $\cM_c$ and $\cA_c$ the corresponding projected sets.


Another interesting characterization of the Aubry set is provided by {\it weak KAM theory} \cite{Fathibook}, in terms of {\it critical subsolutions} of Hamilton-Jacobi equation or, in a more geometric way, in terms of some special {\it Lipschitz Lagrangian graphs}. Let us consider the Hamiltonian system associated to our Tonelli Lagrangian. In fact, if one considers the {Legendre transform} associated to $L$, \ie the diffeomorphism
$\cL_L: \rT M \longrightarrow \rT^*M$ defined by
$\cL_L(x,v)=(x, \frac{\dpr L}{\dpr v}(x,v))$, then it is possible to introduce a {Hamiltonian system} 
$H: \rT^* M \rightarrow \R$, where $H(x,p)=\sup_{v\in \rT_xM}(\langle p,v \rangle - L(x,v))$. It is easy to check that $H$ is also $C^2$, strictly convex and uniformly superlinear in each fiber: $H$ is also said to be {\it Tonelli} (or sometimes {\it optical}). Recall that the flow $\Phi^H_t$ on $\rT^* M$ associated to this Hamiltonian, known as the {\it Hamiltonian flow} of $H$, is conjugated - via the Legendre transform - to the Euler-Lagrange flow of $L$.
Therefore one can define the analogue of the Mather and Aubry sets in the cotangent space, simply considering $\cM^*_c:=\cL_L\big(\widetilde{\cM}_c\big)$ and
$\cA^*_c:=\cL_L\big(\widetilde{\cA}_c\big)$. These sets still satisfy all the properties mentioned above, including the graph theorem.
Moreover, they will be contained in the energy level $\{H(x,p)=\a(c)\}$, where $\a: \rH^1(M;\R)\longrightarrow \R$ is exactly {\it Mather's $\a$-function} introduced before. \\
Using the results in  \cite{Fathibook} it is possible to obtain a nice characterization of the Aubry set. % in terms of {critical subsolutions} of Hamilton-Jacobi equation. 
As above, let $\eta_c$ be a closed $1$-form with cohomology class $c$; we shall say that $u \in C^{1,1}(M)$ is an $\eta_c$-critical subsolution if it satisfies $H(x,\eta_c + d_xu)\leq \a(c)$ for all $x\in M$. The existence of such functions has been showed by P. Bernard \cite{bernardc11}.
If one denotes  by $\cS_{\eta_c}$ the set of $\eta_c$-critical subsolutions, then: 
\beqa{Aubry} {\cA}_c^*=\bigcap_{u\in \cS_{\eta_c}} \left\{(x,\eta_c(x) + d_xu):\; x\in M\right\}.\eeqa
This set does not depend on the particular choice of $\eta_c$, but only on its cohomology class.
%
Observe that since in $(\rT^*M,\omega)$  there is a $1$-$1$ correspondence between Lagrangian graphs and closed $1$-forms, then %(see for instance \cite{AnnaCannas}). 
 we can interpret the graphs of these critical subsolutions as { Lipschitz} Lagrangian graphs in $\rT^*M$ and the Aubry set may be seen as their  {\it non-removable} intersection. 


%Before concluding this section, let us also point out relations between the Aubry sets corresponding to various different cohomology classes. Analogously to what we have seen in Lemma \ref{lemma1}, this relation may be expressed in terms of some geometric property of the $\a$ function, in particular its strictly convexity or better the lack thereof. The regions where $\a$ is not strictly convex are called {\it flats}. For instance the Legendre transform of $\a$ at $c$, denoted $\dpr \a(c)$, which is the set of  homology class for which equality (\ref{fenchelequality}) holds, is an example of flat. It is easy to check that if two cohomology classes are in the relative interior of the same flat $F$ of $\a$, then their Mather sets coincide (see \cite{Mather91}). We denote by $\widetilde{\cM}(F)$ the common Mather set to all the cohomologies in the interior of $F$. 

%In \cite{Massart03} such a result is extended to the Aubry sets.
%\begin{Prop}
%If a cohomology class $c_1$ belongs to a flat $F_c$ of $\a$ containing $c$ in its relative interior, then 
%$\cA_c\subset \cA_{c_1}$. In particular, if $c_1$ lies in the interior of $F_c$ then $\cA_c=\cA_{c_1}$. Conversely, if two cohomology classes $c$ and $c_1$ are such that $\widetilde{\cA}_c \cap \widetilde{\cA}_{c_1} \neq 0$, then $\a$ has a flat containing $c$ and $c_1$.
%\end{Prop}

%So for any flat $F$ of $\a$ and any $c_1,c_2$ in the interior of $F$, the Aubry sets $\widetilde{\cA}_{c_1}$ and $\widetilde{\cA}_{c_2}$ coincide. We denote by $\widetilde{\cA}(F)$ the common Aubry set to all the cohomologies in the interior of $F$.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Main results}

In this section we prove the main results stated in the introduction.
Let $M$ be a closed surface, not necessarily orientable,  $L: \rT M \longrightarrow \R$ a Tonelli Lagrangian on $M$ and $H: \rT^* M \longrightarrow \R$ the associated Hamiltonian. 

Let us recall that a homology class $h$ is said to be $k$-irrational, if $k$ is the dimension of the smallest subspace of $\rH_1(M;\R)$ generated by integer classes and containing $h$. In particular, $1$-{\it irrational} means ``on a line with rational slope'', while {\it completely irrational} stands for $\dim \rH_1(M;\R)$-irrational. Moreover, a homology $h$ is said to be {\it singular} if its Legendre transform $\dpr \beta(h)$ is a {\it singular flat}, \ie its Mather set $\widetilde{\cM}(\dpr \beta(h))$ - see Remark \ref{mathersetsdifferentcohomology} -  contains fixed points. Observe that the set of singular classes, unless it is empty,  contains the zero class and is compact.


For $h \in \rH_1(M;\R)\setminus \left\{0\right\}$, we define the maximal radial flat $R_h$ of $\beta$ containing $h$ as the largest subset of the half-line $\left\{th \; : t \in \left[0,+\infty \right[ \right\}$ containing $h$ (not necessarily in its relative interior) in restriction to which $\beta$ is affine. The possibility of radial flats is the most obvious difference between the $\beta$ functions of Riemannian metrics \cite{Massart97b,nonor} and those of general Lagrangians. An instance of radial flat is found for instance in \cite{Carneiro-Lopes}. We define the Mather set $\widetilde{\mathcal{M}}(R_h)$  as  %the closure in $TM$ of 
the union of the supports of all action minimizing measures with rotation vector $th$, for $th \in R_h$.

Let $h$ be a homology class. Assume $h$ is  1-irrational.
Then for all $t$ such that $th \in R_h$, $th$ is also 1-irrational and $\dpr \beta(th)=\dpr\beta(h)$ (see \cite[Lemma 17]{Massart09}). In particular, $R_h$ is contained in a flat of $\beta$. So Mather's Graph Theorem and   \cite[Lemma C.3]{Massart09} combine to say that $\widetilde{\mathcal{M}}(R_h)$ is a union of pairwise disjoint periodic orbits, or fixed points,  $\gamma_i$, $i \in I$ where $I$ is some set, not necessarily finite. %Consider Mather's $\beta$ function $\beta: \rH_1(M;\R)\longrightarrow \R$ and its convex conjugate $\a: \rH^1(M;\R)\longrightarrow \R$. 

 We shall start with the following proposition, which generalizes a result by Bangert \cite{Bangert} and Mather \cite{Mather90} for twist maps and geodesic flows on the two-torus. 




\begin{Prop}\label{diff1irrat} Let $L$ be a Tonelli Lagrangian on a closed surface $M$ and let  $h \in \rH_1(M;\R)$ be a non-singular $1$-irrational homology class. Then, $\b$ is differentiable at $h$  if and only if $\cM(R_h)=M$. In that case, $\cM (R_h)=M$ is foliated by closed extremals. \end{Prop}

\begin{Proof}
[$\Longleftarrow$] Since $\cM(R_h)=M$, then $\beta$ is differentiable at $h$. In fact, suppose that $c,c'\in \dpr\beta(h)$; then $\widetilde{\cM}(R_h) \subseteq \widetilde{\cM}_c$ and $\widetilde{\cM}(R_h) \subseteq \widetilde{\cM}_{c'}$ (Lemma \ref{lemma1}). From the graph property of the Mather and Aubry sets it follows that  $\widetilde{\cM}_c = \widetilde{\cM}_{c'}$ and $\widetilde{\cA}_c = \widetilde{\cA}_{c'}$. In particular, if 
$\cL_L: \rT M \longrightarrow \rT^*M$ denotes the Legendre transform associated to $L$, 
then  ${\cA}^*_c=\cL_L(\widetilde{\cA}_c)$  and ${\cA}^*_{c'}=\cL_L(\widetilde{\cA}_{c'})$  are invariant Lipschitz Lagrangian graphs and since  they coincide, they must have the same cohomology class, \ie $c=c'$.\\
%Moreover, since each minimizing measure with rotation vector $h$ is supported on a periodic orbit with positive period, then $h$ must be non-zero and $1$-irrational.\\
\noindent [$\Longrightarrow$]
Since $\b$ is differentiable at $h$, let us denote  ${c_h}:=\dpr\beta(h)$.  From \cite[Theorem 3]{Massart09}  it follows that 
$\widetilde{\cM} (R_h) = \widetilde{\cM}_{c_h} = \widetilde{\cA}_{c_h}$ and it is a union of periodic orbits. From this, one can deduce that:
\begin{itemize}
%\item  $\a$ is differentiable at $c$ and $\dpr \a(c)=h$. In fact, if $h'\in \dpr \a(c)$, then 
%$\widetilde{\cM}^{h'} \subseteq \widetilde{\cM}_c=\widetilde{\cM}^h$. This is not possible, since all measures in $\widetilde{\cM^h}$ are supported on periodic orbits with homology $h$
\item for all $c'\neq {c_h}$, $\widetilde{\cM}_{c_h} \cap \widetilde{\cM}_{c'}= \emptyset$; otherwise, $\widetilde{\cM} (R_h)\cap \widetilde{\cM}_{c'} \neq \emptyset$ and this would imply that $c'\in \dpr \b(h)$ (Lemma \ref{lemma1}), contradicting the differentiability of $\b$ at $h$.
\item $\a$ is strictly convex at ${c_h}$: it is an easy consequence of the disjointness of  $\widetilde{\cM}_{c_h}$ from the other  Mather sets and Remark \ref{mathersetsdifferentcohomology}.
%In fact, suppose by contradiction that there exist $\l\in (0,1)$ and $c,c'\in \rH^1(M;\R)$ such that
%$\a(\l c + (1-\l)c') = \l \a(c) + (1-\l)\a(c')$ and let $\m_{\l}$ be a $(\l c + (1-\l)c')$ - action minimizing measure. Let us denote by $\eta_c$ and $\eta_{c'}$ two closed $1$-forms with cohomology classes, respectively, $c$ and $c'$. Then:
%\beqano
%-\a(\l c + (1-\l)c') &=& \int \big(L - \l \eta_c - (1-\l)\eta_{c'} \big) d\m_{\l} = \\
%&=& \l \int \big(L - \eta_c\big) d\m_{\l}  + (1-\l) \int \big(L - \eta_{c'} \big) d\m_{\l}  \geq \\
%&\geq& - \l \a(c) - (1-\l)\a(c') = -\a(\l c + (1-\l)c'),
%\eeqano
%therefore all above inequalities are equalities and this implies that $\m_{\l}$ is also 
%$c$-action minimizing and $c'$-action minimizing.
\end{itemize}

In order to complete the proof, we need to show that $\cA_{c_h}=M$. If this were not true, then, there would exist an open region $\G$ in the complement of the $\cA_{c_h}$, whose boundary consists of two (minimizing)  periodic orbits. $\G$ would be homologically non trivial, therefore one could find a closed non-exact $1$-form on $M$ whose support is disjoint from $\cA_{c_h}$. 
%Using the notation of \cite[pages 2--3]{Massart03}, one can show that $\dim E_c \geq 1$. 
%, or quivalently that  $V_c$ is the (integer) subspace generated by $h$. Therefore $\dim {V_c}^{\perp} =1$.
From \cite[Theorem 1]{Massart03} (see \cite[Theorem 2]{lowdim} for the non-orientable case), this would imply that  ${c_h}$ is contained in the interior of a non-trivial flat of $\a$, contradicting the strict convexity of $\a$ at ${c_h}$. %(strict convexity implies that the maximal face containing $c$ is trivial, \ie, $F_c=\{c\}$).
\end{Proof}


\begin{Rem}
Observe that the above result is not true if $h$ is singular. Indeed, take a vector field $X$ on a closed surface $\Sigma_2$ of genus $2$, which consists of periodic orbits, except for a singular graph with two fixed points and four hetero/homoclinic orbits between the two fixed points (see figure 1).
One can embed it into the Euler-Lagrange flow of a Tonelli Lagrangian, given by 
$L_X(x,v)=\frac{1}{2}\|v-X(x)\|^2_x$. It is possible to show (see for instance \cite{FFR}) that ${\rm Graph}(X)$ is invariant and that in this case $\widetilde{\cA}_0={\rm Graph}(X)$; therefore, the projected Aubry set $\cA_0$ is the whole surface $\Sigma_2$,
%{. I cannot draw so I'll try to describe it : take a surface of genus two. Draw a separating closed curve in the middle. Put your two fixed points on two antipodal points of this separating curve. Then the two halves of the curve give you two homoclinic orbits. Draw two other  homoclinics that go through the holes of the surface. The rest of the surface may now be foliated by closed curves.
%In this case the projected Aubry set  is the whole surface (suppose, up to a modification of the Lagrangian, that we are considering the zero cohomology class) and 
so  $\beta$ is differentiable at all homology classes in $\dpr \a (0)$ (see \cite[Theorem 1]{Massart03}). However,  $\cA_0=\Sigma_2$ is not foliated by closed extremals.
\begin{figure} [h!]
\begin{center}
\includegraphics[scale=3.5]{figure}
\caption{An example of singular Aubry set.}
\end{center}
\end{figure}
\end{Rem}


For the sake of completeness, let us also recall that:

\begin{Prop}\label{prop2}
$\b$ is always differentiable at completely irrational homology classes.
\end{Prop}

\noindent It follows from \cite[Corollary 3]{Massart03}  (see \cite[Theorem 2]{lowdim} for the non-orientable case).\\

%\begin{Prop}
%If $\dim H_1(M;\R)=1$, then $\b$ is always $C^1$ except possibly at the origin.
%{\bf Can we say something more when it is $C^1$ everywhere?}
%\end{Prop}

%\begin{Proof}
%The result it follows from \cite[Corollary 3]{Massart03} (or Carneiro?). {\bf Non-orientability is an issue???}
%\end{Proof}
Let us now recall the definition of $C^0$-integrability (see also \cite{Arnaud}).

\begin{Def}\label{C0integrability}
A Tonelli Hamiltonian $H:\rT^*M \longrightarrow \R$ is said to be $C^0$-integrable, if there exists a foliation of $\rT^* M$ made of invariant Lipschitz Lagrangian graphs, one for each cohomology class.
\end{Def}


\begin{Lem}\label{C_zero_int_beta_C1}
Let $M$ be a compact manifold of any dimension, $L: \rT M \longrightarrow \R$ a Tonelli Lagrangian and $H: \rT^* M \longrightarrow \R$ the associated Hamiltonian. If $H$ is $C^0$-integrable, then $\beta$ is $C^1$.
\end{Lem}

\begin{Proof}
Suppose that $H$ is $C^0$-integrable. This means that the cotangent space $\rT^*M$ is foliated by disjoint invariant Lipschitz Lagrangian graphs. Let us denote  by $\L_c$ the invariant Lagrangian graph of such a foliation, corresponding to the cohomology class $c$. It is easy to check that each $\L_c$ is the graph of a solution of Hamilton-Jacobi equation $H(x,\eta_c+du)=\a(c)$, where $\eta_c$ is a closed $1$-form on $M$ with cohomology class $c$,
 and therefore
from weak KAM theory \cite{Fathibook} it follows that for each $c\in \rH^1(M;\R)$ the Aubry set $\cA^*_c \subseteq \L_c$.
If for some $h\in\rH_1(M;\R)$ there exist $c\neq c'$ such that $c,c'\in\dpr \b(h)$, then  $\widetilde{\cM}^h \subseteq \widetilde{\cM}_c\cap \widetilde{\cM}_{c'}$ (Lemma \ref{lemma1}). But this implies that $\widetilde{\cM}_c\cap \widetilde{\cM}_{c'} \neq 0$ and therefore 
$
\L_c \cap \L_{c'} \supseteq \cA_c^* \cap \cA^*_{c'} \neq 0,
$
which contradicts the disjointness of the Lagrangian graphs forming the foliation. Hence, $\b$ must be differentiable everywhere.\\
\end{Proof}

We can now prove the main results stated in the Section \ref{section1}.\\

\begin{Lem}\label{maintheo2}
Let $M$ be a closed surface other than the sphere, the projective plane, the Klein bottle or the torus,  and let $L:\rT M \longrightarrow \R$ be a Tonelli Lagrangian. Then, $\beta$ is not differentiable at any 1-irrational non-singular homology class.
\end{Lem}

\begin{Proof}
 If $\beta$  is differentiable at some non-singular $1$-irrational homology class, from Proposition \ref{diff1irrat} it follows that  the Mather set corresponding to such a class projects over the whole manifold and consists of periodic orbits. But this leads to a contradiction, since it implies the existence of a fixed-point-free vector field on $M$.
 \end{Proof}
 
 When $\beta$ is $C^1$ everywhere, we can improve on Proposition \ref{diff1irrat}, by ruling out radial flats of $\beta$.


 \begin{Prop} \label{C1_implies_convex}
Let $L$ be an autonomous Tonelli Lagrangian on the two-torus such that $\beta$ is differentiable at every point of $\rH_1(\T^2;\R)$. Then for all $h \in \rH_1(\T^2;\R) \setminus \{ 0 \}$, we have $R_h = \{ h \}$. In particular, $\beta$ is strictly convex.
\end{Prop}
\begin{Proof} 

{\bf First case}: $h$ is 1-irrational.

Replacing, if we have to,  $h$ with an extremal point of $R_h$, which is also 1-irrational, we may assume that $R_h = \left[ \lambda h, h \right]$ for some $0 \leq \lambda \leq 1$. We want to prove that $\lambda =1$. Pick:
 \begin{itemize}
  \item $(x,v) \in \rT\T^2$ such that the probability measure carried by the orbit $\phi_t (x,v) $ has homology $\lambda h$,
    \item a sequence of real numbers $t_n \geq 1$ such that $t_n$ converge to $1$
  \item for each $n \in \N$, $h_n:= t_n h$ and   $c_n \in \rH^1 (\T^2;\R)$ such that $\partial \beta (h_n) =c_n$ (recall that $\beta$ is differentiable at $h_n$).
  \end{itemize}
Then by Proposition \ref{diff1irrat}, since $\beta$ is differentiable at $t_nh$, which is 1-irrational, we have $\cM_{c_n}=M$; thus for every $n$ there exists a $v_n \in T_x M$ such that $(x,v_n) \in  \widetilde{\cM}_{c_n} $, and moreover the orbit $\phi_t (x,v_n) $ is periodic. By semicontinuity of the Aubry set in dimension two \cite{Bernard08} and by the Graph Theorem, $v_n$ converges to $v$ when $n$ goes to infinity (in fact $c_n$ converges to $c:=\dpr\beta(R_h)$). Let $T$ and $T_n$ be the minimal periods of 
$\phi_t (x,v) $ and $\phi_t (x,v_n) $, respectively. If $(x,v)$ is a fixed point we just set $T := + \infty$. We now prove that $\liminf T_n \geq T$.
%
Indeed, if  some subsequence $T_{n_k}:= T_k$ converged to $T' < T$, we would have 
$\phi_{T' }(x,v) = (x,v)$, contradicting the minimality of $T$. If $(x,v)$ is a fixed point, we have $v=0$, so $v_n$ converges to zero, so $T_n$ tends to infinity. 


Let $h_0 \in \rH_1(\T^2;\Z)$ be a generator of $\R h \cap \rH_1(\T^2;\Z)$ such that the probability measures carried by the orbits $\phi_t (x,v) $ and $\phi_t (x,v_n) $ have homologies $T^{-1} h_0 = \lambda h$ and $T_n ^{-1} h_0$, respectively. 

Now  if 
$\lambda< 1$, since $\liminf T_n \geq T$, for $n$ large enough there would exist a   $c_n$-minimizing measure with homology in $\left[ \lambda h, h \right)$. This means that the radial flats $R_h$ and $R_{t_n h}$ overlap, in other words, 
$t_n h \in R_h$. This contradicts the fact that $R_h = \left[ \lambda h, h \right]$, hence 
$\lambda =1$.

{\bf Second case}: $h$ is 2-irrational, that is, completely irrational. 

Then any action minimizing measure with rotation vector  $h$ is supported on a lamination of the torus without closed leaves. Any such lamination is uniquely ergodic, in particular $h$ is not contained in any non-trivial flat of $\beta$, radial or not,  regardless of the Lagrangian.

The statement about the differentiability of $\beta$ implying its strict convexity is now just a consequence of the fact, observed by M. Dias Carneiro in \cite{Carneiro}, that for an autonomous Lagrangian on a closed, orientable surface $M$, the flats of $\beta$ are contained in isotropic subspaces of $\rH_1 (M;\R)$ with respect to the intersection symplectic form on $\rH_1 (M;\R)$. In particular, when $M = \T^2$, all flats are radial.
 \end{Proof}
 
 \begin{Lem}\label{lemma0}
Let  $L: \rT \T^2 \longrightarrow \R$ be a Tonelli Lagrangian. Assume that   $\beta$ is $C^1$ everywhere.
Then zero is  (possibly) the only  singular class, and for every 1-irrational, nonzero homology class $h$, $\cM^h = \T^2$, and $\cM^h$ is foliated by periodic orbits with the same homology class and minimal period. 
\end{Lem}
\begin{Proof}
First note that if a non-zero class $h$ is singular, then the Mather set of $R_h$ contains a fixed point, so $R_h$ contains the homology of the Dirac measure on the fixed point, which is zero. This contradicts Proposition \ref{C1_implies_convex}, which says that $R_h = \left\{ h \right\}$.

Now take a non-zero, 1-irrational homology class $h$; so $h$ is non-singular and by Proposition \ref{diff1irrat}, $\cM (R_h)=\T^2$ and $\cM (R_h)$ is foliated by periodic orbits. Since  $R_h = \left\{ h \right\}$ by Proposition \ref{C1_implies_convex}, we have $\cM^h = \T^2$. It is well known that all leaves of a foliation of the two-torus by homologically non trivial closed curves are homologous. So there exists $h_0 \in \rH_1 (\T^2;\Z)$ such that the projection to $\T^2$ of any orbit contained in $\cM^h$ is homologous to $h_0$. Besides, for each $x \in \T^2$, let $T(x)$ be the minimal period of the periodic orbit $\gamma_x$ in $\cM^h$ through $x$. The homology class of the probability measure carried by $\gamma_x$ is 
$T^{-1}h_0$. Now the fact that there are no non-trivial radial flats of $\beta$ implies that $T(x)$ is independent of $x$.
\end{Proof}

\begin{Prop} \label{torus_only_integrable}
The torus is the only closed surface which admits a $C^0$-integrable Hamiltonian.
\end{Prop}
\begin{Proof}

First,  no Hamiltonian on the sphere can be $C^0$-integrable. Indeed, any Lagrangian graph is exact since the sphere is simply connected, and any two exact Lagrangian graphs intersect, because any $C^1$ function on the sphere has critical points.

Likewise, no Hamiltonian on the projective plane can be 
$C^0$-integrable, for its lift to the sphere would be $C^0$-integrable.

For the Klein bottle $\K$, we need to work a little bit more. 
 For each $x \in \K$, let us define
\beqano
F_{x} : \rH^1(\K;\R)\simeq \R &\longrightarrow & \rT^*_x \K \simeq \R^2\\
c &\longmapsto& \L_c (x),
\eeqano
where $\L_c$, for  $c\in\rH^1(\K;\R)$, are the Lagrangian graphs foliating $\rT^*\K$.
It is easy to check that $F_{x}$ is continuous and  injective (as it follows from the disjointness of the $\L_c$'s). Moreover, if the Hamiltonian is $C^0$-integrable, the map $F_x$ is surjective. Now there is no such thing as a continuous bijection from $\R$ to $\R^2$, so there is no $C^0$-integrable Hamiltonian on the Klein bottle. 

The same argument can be used for any surface with first Betti number $>2$.

 Finally,
no Hamiltonian on the connected sum of three projective planes (first Betti number equal to $2$) can be $C^0$-integrable, for it would  lift to  a $C^0$-integrable Hamiltonian on a surface of genus two.




\end{Proof}

\begin{Teo}\label{maintheo1}
Let  $L:\rT \T^2 \longrightarrow \R$ be a Tonelli Lagrangian on the two-torus. 
Then, $\beta$ is $C^1$ if and only if the system is $C^0$-integrable.
\end{Teo}

\begin{Proof}
[$\Longleftarrow$] It follows from Lemma \ref{C_zero_int_beta_C1}. 
 [$\Longrightarrow$] For each homology class $h$, let us denote  $c_h:=\dpr \beta(h)$. 
% Observe that since $\beta$ is differentiable everywhere, then if $h\neq 0$ then it is non-singular. 
 If $h$ is non-singular and $1$-irrational, then $\L_{c_h}:= \cA^*_{c_h}$ is an invariant Lipschitz Lagrangian graph of cohomology class $c_h$, which is foliated by periodic orbits of homology $h$. 
 
 Observe that such cohomology classes $c_h$ are dense in $\rH^1(M;\R)$. Indeed, 1-irrational non-singular homology classes are dense in $\rH_1(M;\R)$ 
 because 1-irrational  homology classes are dense in $\rH_1(M;\R)$ and, by Lemma \ref{lemma0}, zero is the  
 only possibly singular one.
 On the other hand, since $\beta$ is differentiable, the Legendre transform is a homeomorphism from $\rH_1(M;\R)$ to $\rH^1(M;\R)$. 
 
 
   Using the semicontinuity of the Aubry set in dimension $2$ (see \cite{Bernard08}), we can deduce that for each $c\in \rH^1(M;\R)$ the Aubry set $\cA^*_c$ projects over the whole manifold and therefore it is still an invariant Lipschitz Lagrangian graph, which we shall denote $\L_c$. Observe that all these $\L_c$'s are disjoint. This is a straightforward consequence of the differentiability of $\beta$. In fact if for some $c\neq c'$ we had that $\L_c \cap \L_{c'} \neq 0$, then  $\widetilde{\cM}_c\cap \widetilde{\cM}_{c'} \neq 0$; but this would contradict the differentiability of $\beta$ at some homology class. It only remains to prove that the  union is the whole $\rT^*M$. Observe that for each $c\in \rH^1(M;\R)$, $\L_c$ is the graph of the unique solution of the Hamilton-Jacobi equation $H(x,\eta_c + du_c) = \a(c)$, where $u_c\in C^{1,1}(M)$ and $\eta_c$ is a closed $1$-form on $M$ with cohomology class $c$.  For each ${x_0}\in M$, let us define
\beqano
F_{x_0} : \rH^1(M;\R)\simeq \R^2 &\longrightarrow & \rT^*_{x_0}M \simeq \R^2\\
c &\longmapsto& \eta_c + d_{x_0}u_c.
\eeqano
It is easy to check that $F_{x_0}$ is continuous (actually locally Lipschitz); moreover, it is injective (as it follows from the disjointness of the $\L_c$'s). Therefore, $F_{x_0}(\R^2)$ is open \cite{Brouwer}. Using Ascoli-Arzel\`a's theorem, one can show that this image is also closed. In fact, let $y_k = F_{x_0}(c_k)$ be a sequence in $\rT^*_x M$ converging to some $y_0$. Since each family of classical solutions of Hamilton-Jacobi on which the $\a$-function (\ie the energy) is bounded gives rise to a family of functions with uniformly bounded  Lipschitz constants, then there exist
$\tilde{u} \in C^{1,1}(M)$ and $\tilde{c} \in \rH^1(M;\R)$, such that 
$c_k+du_{c_k} \rightarrow \tilde{c}+d\tilde{u}$ uniformly. From the continuity of the $\a$-function, it follows that $H(x,\tilde{c}+d_x\tilde{u})=\a(\tilde{c})$. Since $\L_{\tilde{c}}$ is the unique invariant Lagrangian graph with cohomology $\tilde{c}$ (essentially because it coincides with the Aubry set, see \cite{Fathibook}), then $\L_{\tilde{c}}=\{(x,\tilde{c}+d_x\tilde{u}:\; x\in M\}$ and therefore $y_0 = \tilde{c} + d_{x_0}\tilde{u} = F_{x_0}(\tilde{c}) \in F_{x_0}(\R^2)$. This shows that $F_{x_0}(\R^2)$ is also closed and therefore it is all of $\R^2$. Since this holds for all $x_0\in M$, it follows that $\cup_c \L_c = \rT^*M$. 
\end{Proof}

We can deduce more information about the dynamics on a $C^0$-integrable system.

\begin{Cor}\label{corollarymaintheo}
Let $H: \rT^* M \longrightarrow \R$ be a $C^0$-integrable Hamiltonian on a two-dimensional closed manifold $M$. Then, $M$ is diffeomorphic to $\T^2$. Moreover:
\begin{itemize}
\item[{\rm (}i{\rm)}]  for each $1$-irrational homology class $h$, there exists an invariant Lagrangian graph foliated by periodic orbits with the same homology and minimal period;
\item[{\rm (}ii{\rm)}] for each completely irrational homology class, there exist an invariant Lagrangian graph on which the motion is conjugated to an irrational rotation on the torus or to a Denjoy type homeomorphism; 
\item[{\rm (}iii{\rm)}] there exists a dense $G_{\delta}$ set of (co)homology classes, for which the motion on the corresponding invariant torus is conjugated to a rotation;
\item[{\rm (}iv{\rm)}] as for the $0$-homology class, there exists a $C^1$ invariant torus  $\L_{c(0)}= \{(x,\frac{\dpr L}{\dpr v}(x,0):\; x\in \T^2\}$ consisting of fixed points.
\end{itemize}
\end{Cor}



\begin{Proof}
If $H$ is $C^0$ integrable, then $\b$ is $C^1$ (Lemma \ref{C_zero_int_beta_C1}). By Proposition \ref{torus_only_integrable}, $M$ is diffeomorphic to the 2-torus.
%Observe that $M$ cannot be non-orientable. In fact, if $M$ were the connected sum of three projective planes, then there could not be a foliation of $M$ by closed curves, because it would lift to a foliation of the genus $2$ orientation cover by closed curves.\\

\noindent
({\it i})  It is a consequence of Lemma \ref{lemma0}.
\noindent ({\it ii}) Let $h$ be a completely irrational homology class and let $c_h=\dpr \beta(h)$. We shall follow the discussions in \cite[p.~1084]{Carneiro} (see also \cite[p.~66 et seqq.]{Mather09} for more technical details). The Hamiltonian flow on $\L_{c_h}$ induces a flow on $\T^2$ without fixed points or closed trajectories. Thus, one can construct a non-contractible closed curve $\Gamma$, which is  transverse to the flow. Using an argument {\it \`a la} Poincar\'e-Bendixon, it follows that all trajectories starting on $\Gamma$ must return to it. Hence,  one  can define a continuous map from a compact subset of $\Gamma$ to itself, which is order preserving and with irrational rotation number. Therefore, it is either conjugate to an irrational rotation or a Denjoy type homeomorphism.\\
\noindent({\it iii}) From ii), it is sufficient to show that the set of homology classes for which the Mather set projects over the whole torus is a dense $G_{\delta}$ set; in fact if this is the case, the flow cannot be conjugated to a Denjoy type homeomorphism. Let us start by observing that this set is clearly dense since it contains all $1$-irrational homology classes. The fact that it is also $G_{\delta}$ follows from \cite[Corollaire 4.5]{FathiHerman}, in which it is proven that 
 the set of strictly ergodic flows on a compact set is $G_{\delta}$ in the $C^0$ topology.
 %
 One could also observe that since our Lagrangian is $C^0$-integrable, by \cite[Th\'eor\`eme 3]{Arnaud}, there is a dense $G_{\delta}$ of cohomology classes with a $C^2$ weak KAM solution; and by a recent result of Fathi \cite{Fathi_Kyoto}, when the flow is of Denjoy type, there cannot be a $C^2$ weak KAM solution.  \\
\noindent({\it iv}) Since the union of the Aubry sets foliates all the phase space, then any point $(x,0)$ will
be contained in some Aubry set and therefore, using  \cite[Proposition 3.2]{FFR}, it follows that 
it is a fixed point of the Euler-Lagrange flow. Hence, the Dirac-measure $\d_{(x,0)}$ is invariant and action minimizing (since its support is contained in some Aubry set); clearly, such a measure  has rotation vector equal to $0$. Therefore, $\widetilde{\cM}^0 \supseteq \T^2\times \{0\}$ and from the graph property it follows that they coincide. Then, 
$\Lambda_{c(0)}:=\cA^*_{c(0)} = \left\{(x,\frac{\dpr L}{\dpr v}(x,0):\; x\in \T^2 \right\}$, which is at least $C^1$.\\
\end{Proof}


\begin{Rem}
Although we believe that the motions on all invariant Lagrangian tori must be conjugated to rotations, we have not been able to show more than $iii)$.
It is not clear to us  whether it is possible or not that a Denjoy type homeomorphism is embedded into a $C^0$-integrable Hamiltonian system.
\end{Rem}

\subsection{The case of mechanical Lagrangians}\label{mechanicalcase}
Recall that a mechanical  Lagrangian is of the form $L(x,v) = 1/2 \,  g_x(v,v) +f(x)$, where $g$ is a Riemannian metric and $f$ is a $C^2$ function on $\T^2$. In this case we can bridge the gap between $C^0$-integrability and complete integrability (in the sense of Liouville), using Burago and Ivanov's theorem on metrics without conjugate points \cite{BI}.
 
\begin{Prop}\label{prop_mechanical}
Let $L$ be a $C^0$-integrable mechanical Lagrangian on an $n$-dimensional torus. Then the potential $f$ is identically zero and the metric $g$ is flat. In particular, $L$ is completely integrable.
\end{Prop}

\begin{Prop}\label{prop_mechanical_2dim}
Let $L$ be a  mechanical Lagrangian on a $2$-dimensional torus, whose $\beta$-function is $C^1$. Then the potential $f$ is identically zero and the metric $g$ is flat. In particular, $L$ is completely integrable.
\end{Prop}

First we need the following lemma (see also \cite[Lemma 1]{Sorrentino08}).

\begin{Lem}
Let\begin{itemize}
  \item $M$ be a closed manifold of any dimension
  \item $L$ be an autonomous Tonelli Lagrangian on $M$, such that $L(x,v)=L(x,-v)$ for all $(x,v)$ in $TM$. 
\end{itemize}
Then, the Aubry set $\widetilde{\cA}_0$ consists of fixed points of the Euler-Lagrange flow.
\end{Lem}
\begin{Proof}
Take $(x,v) \in \widetilde{\cA}_0$. Then, by \cite{Fathibook} there exists a sequence of $C^1$ curves $\gamma_n \; : \; \left[0, T_n \right] \longrightarrow M$, such that 
\begin{itemize}
  \item $\gamma_n (0) = \gamma_n (T_n) = x$ for all $n\in \N$
  \item $T_n \longrightarrow +\infty$ as $n\longrightarrow +\infty$
   \item $\dot{\gamma}_n (0) \longrightarrow v$ and $\dot{\gamma}_n (T_n) \longrightarrow v$ as $n\longrightarrow +\infty$
   \item $\int_{0}^{T_n}\left( L(\gamma_n , \dot{\gamma}_n) + \alpha(0)\right)dt \longrightarrow 0$    as $n\longrightarrow +\infty$.   
   \end{itemize}
   Consider the sequence of curves $\delta_n \; : \; \left[0, T_n \right] \longrightarrow M$, such that $\delta_n (t) := \gamma_n (T_n -t)$ for all $n \in \N$, $t \in \left[0, T_n \right]$.
   Then, since $L$ is symmetrical, $\int_{0}^{T_n} ( L(\delta_n , \dot{\delta}_n) + \a(0) ) dt \longrightarrow 0$    as $n\longrightarrow +\infty$, which proves that  $(x,-v) \in \widetilde{\mathcal{A}}_0(L)$.  Therefore, by Mather's Graph Theorem, $v=0$, so $(x,v)$ lies in the zero section of $TM$. Now, by \cite[Proposition 3.2]{FFR}, for any Lagrangian, the intersection of $\widetilde{\cA}_0$ with    the zero section of $TM$ consists of fixed points of the Euler-Lagrange flow. This proves the lemma. 
   \end{Proof}
   
  Now let us assume that the Lagrangian is mechanical. Then, the only fixed points of the Euler-Lagrange flow are the critical points of the potential $f$, and the only minimizing fixed points are the minima of $f$. So if $f$ is not constant, the Lagrangian cannot be $C^0$-integrable. Furthermore, since the Lagrangian is $C^0$-integrable, every orbit is minimizing, in particular, there are no conjugate points. So by Burago and Ivanov's proof of the Hopf Conjecture \cite{BI}, the metric $g$ is flat. This proves Proposition \ref{prop_mechanical}.  Proposition \ref{prop_mechanical_2dim} is now just a consequence of Theorem    \ref{maintheo1}.
   
   
     

%Let $k:= \dim H_1(M;\R) >2$. Suppose by contradiction that $\b$ is differentiable on an open subset $U\subseteq \rH_1(M;\R) \simeq \R^k$. Proceeding as before, one can show that for each $h \in \overline{U}$, $\L_{c_h}:= \cA^*_{c_h}$ is an invariant Lagrangian graph of cohomology class $c(h)$. The set of such cohomology classes, which we shall denote by $V$, is a closed set with non-empty interior. Since $\b$ is differentiable on $U$, then these Lagrangian graphs are disjoint (same proof as in Proposition \ref{proptorus}). Let us now fix $x_0\in  M$ and consider the function $F_{x_0} : V  \longrightarrow  \rT*_xM \simeq \R^2$ defined as in Proposition \ref{proptorus}. This function is continous and injective, but this is impossible since $k > 2$.





%%%%%%%%%%%%%%%%%%%REFERENCES %%%%%%%%%%%%%%%%%%%
\def\cprime{$'$}
\begin{thebibliography}{10}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}


\bibitem{Arnaud}
Marie-Claude Arnaud.
\newblock Fibr\'es de Green et r\'egularit\'e des graphes $C^0$-Lagrangiens invariants par un flot de Tonelli.
\newblock {\em Ann. Henri Poincar\'e}, 9 (5): 881--926, 2008.


\bibitem{Bangert}
   Victor Bangert.
   \newblock Geodesic rays, Busemann functions and monotone twist maps.
   \newblock {\em Calc. Var. Partial Differential Equations}, 2 (1): 49--63, 1994.

\bibitem{nonor}
Florent Balacheff and Daniel Massart.
\newblock Stable norms of non-orientable surfaces.  
\newblock {\em Ann. Inst. Fourier (Grenoble)}  58 (4):  1337--1369, 2008.

\bibitem{bernardc11}
Patrick Bernard.
\newblock Existence of $C^{1,1}$ critical subsolutions of the Hamilton-Jacobi equation on compact manifolds.
\newblock {\em Ann. Sci. \'Ecole Norm. Sup.}, 40 (3): 445--452, 2007.

\bibitem{Bernard08}
Patrick Bernard.
\newblock On the Conley decomposition of Mather sets.
\newblock {\em Rev. Mat. Iberoamericana}, to appear.


\bibitem{Brouwer}
Luitzen~E.~J.  Brouwer.
\newblock Zur invarianz des $n$-dimensionalen Gebiets.
\newblock {\em Math. Ann.}, 72 (1): 55--56, 1912.


\bibitem{BI}
Dmitri Burago and  Sergey Ivanov.
\newblock Riemannian tori without conjugate points are flat.  
\newblock {\em Geom. Funct. Anal},  4  (3): 259--269, 1994.

\bibitem{Carneiro}
Mario J. Dias Carneiro.
\newblock On minimizing measures of the action of autonomous {L}agrangians.
\newblock {\em Nonlinearity}, 8 (6): 1077--1085, 1995. 


\bibitem{Carneiro-Lopes}
Mario J. Dias Carneiro and Arthur  Lopes.
\newblock On the minimal action function of autonomous Lagrangians associated to magnetic fields.  
\newblock {\em Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire},  16 (6): 667--690, 1999.

\bibitem{FathiHerman}
Albert Fathi and Michael~R. Herman.
\newblock Existence de diff\'eomorphismes minimaux.
\newblock  {\em Ast\'erisque} 49: 37--59, 1977. 

\bibitem{Fathibook}
Albert Fathi.
\newblock The Weak {K}{A}{M} theorem in {L}agrangian dynamics.
\newblock Cambridge University Press (to appear).

\bibitem{Fathi_Kyoto}
Albert Fathi.
\newblock {D}enjoy-{S}chwartz and {H}amilton-{J}acobi.
\newblock {\em RIMS meeting: Viscosity solutions of differential equations and related topics}, Kyoto University, 2008.

\bibitem{FFR}
Albert Fathi, Alessio Figalli, and Ludovic Rifford 
\newblock On the Hausdorff dimension of the Mather quotient.  
\newblock {\em Comm. Pure Appl. Math.},  62 (4): 445--500, 2009.



\bibitem{Massart97b}
Daniel Massart.
\newblock Stable norms of surfaces: local structure of the unit ball of rational
 directions.
\newblock {\em Geom. Funct. Anal.}, 7 (6): 996--1010, 1997.

\bibitem{Massart03}
Daniel Massart.
\newblock On Aubry sets and Mather's action functional.
\newblock {\em Israel J. Math.}, 134: 157--171, 2003.

\bibitem{Massart09}
Daniel Massart.
\newblock Aubry sets vs Mather sets in two degrees of freedom.
\newblock Preprint, 2009.

\bibitem{lowdim}
Daniel Massart.
\newblock Differentiability of Mather's beta function in low dimensions.
\newblock Preprint, 2009.

\bibitem{Mather90}
John~N. Mather.
\newblock Differentiability of the minimal average action as a function of the rotation number.
\newblock {\em Bol. Soc. Brasil. Mat. (N.S.)}, 21 (1):  59--70, 1990.
		

\bibitem{Mather91}
John~N. Mather.
\newblock Action minimizing invariant measures for positive definite
{L}agrangian systems.
\newblock {\em Math. Z.}, 207 (2): 169--207, 1991.

\bibitem{Mather93}
John~N. Mather.
\newblock Variational construction of connecting orbits.
\newblock {\em Ann. Inst. Fourier (Grenoble)}, 43 (5): 1349--1386, 1993.


\bibitem{Mather09}
John~N. Mather.
\newblock Order structure on action minimizing orbits.
\newblock Preprint, 2009.

\bibitem{Sorrentino08}
Alfonso Sorrentino.
\newblock On the total disconnectedness of the quotient Aubry set.
\newblock {\em Ergodic Theory Dynam. Systems}, 28 (1): 267--290, 2008.





\end{thebibliography}

\vspace{1.truecm}

\noindent {\sc Daniel Massart}, 
{\it D\'epartement de Math\'ematiques,
Universit\'e Montpellier 2, France.
\\
email: {\sc massart@math.univ-montp2.fr}}\\



\noindent {\sc Alfonso Sorrentino}, { \it CEREMADE, UMR CNRS 7534,
Universit\'e Paris-Dauphine, France
\\
email: {\sc alfonso@ceremade.dauphine.fr}}\\

\end{document}
---------------0907121236356
Content-Type: application/postscript; name="figure.eps"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="figure.eps"

%!PS-Adobe-3.0 EPSF-3.0
%%Creator: GIMP PostScript file plugin V 1,17 by Peter Kirchgessner
%%Title: 548px-Double_torus_illustration.eps
%%CreationDate: Wed Jun 10 15:10:52 2009
%%DocumentData: Clean7Bit
%%LanguageLevel: 2
%%Pages: 1
%%BoundingBox: 14 14 60 56
%%EndComments
%%BeginProlog
% Use own dictionary to avoid conflicts
10 dict begin
%%EndProlog
%%Page: 1 1
% Translate for offset
14.173228346456694 14.173228346456694 translate
% Translate to begin of first scanline
0 41.466816647919003 translate
45.354330708661415 -41.466816647919003 scale
% Image geometry
315 288 8
% Transformation matrix
[ 315 0 0 288 0 0 ]
currentfile /ASCII85Decode filter /RunLengthDecode filter
%%BeginData:        53983 ASCII Bytes
image
JcC<$`rCViJcEaiJcC<$`rCViJcEaiJcC<$`rCViJcEaiJcC<$`rCViVuI#$r;$!OiS2`Wc3VbM
hW=8"q>C6#s+13Ors\o(p[$YPYFUSrBOP$@<Xi,=@r-^VT!$/inG3"]]DmH^Yl>(0qX)qLQ$?G&
4Z`R&p,`.4#s(:B@=Y,nh=CKXs+13VrsA]#lcc8U?VE[Hk;il26;_9P\_ZfR_>f)d[JpL1n^jIV
;*mA-3r(iB4\/n3[GU]W`;bDg\Glg5p"HNj;a<G23r_8F4?Ykm5XGB9r^6]P#sgg@5=Sh0_WLa8
s+13\rrr5RSS(AQloP/.#s:F;6:=7484cGQ9E%Tn91hfI7RTR0<.)IHrQ,"is1J@9rq4m+><b4:
3Y=G:5!VG&7n?6G:/=\`;H$Oor)F2*;c6Ff9h@lQF/dphrlb=ms1SF8pXtp@53t<D4$>eq77TsC
:/=_c<*!+(>$>-8qcsY9>?Y02<DlUd:2?omqoo(ks1eR:r8X>P54(BK4$5\m6q0aA:/Onh='8a4
?=.)K@Uiq[AS1mPs'uOI@:*;E=B/0pFgLK6cMrIq_>a]<n%Ng3475-I4$5\n77U!F:f:7o=Bf'<
@:EbYAnYmnCMWc_%r5ooB4b[`?X-]5?AZhurm:[rs2+d<pst$gm5c7L4?l/$8P;fU<**4,?!h&O
AnYmnCi+*0EH6)@F8g:\Et<;2E,TW1BkLsa>ZP=+iVV%&JcE[g"T@\dAO#QC*]uVP6qBsG;,gV"
>[CiKAS>goD/XB6F*2YMG^+LZrHeHg&9rDBF`VSAD/*]m?=/2ooD[5=JcE^h"T6u08NZ?$+?M_O
6UsaC:fLM"?!h&OB4u$sDfKi?Fa&+WH[L6iIXh6F&q>@ZI=-BeGBIqFChR<bBW:7qeGk+"`r?/1
[UVMg3[-[N6:ORA;,gV#?!h)QBPM@%EH?8HH$XjdIt3-&KS5&6LAco,L(T,mK7\W'I<p-\Ec5](
A;/P.ec14#a8Z81UJj;B3[-^P6qBsG;H6h'?X[JYC2Im/FEVnTH[UBpJqSi4LPUeEMZ/M6N;\YI
MM[1FL4t82IslWeF)Pc(JAq?8s+13lrr`/QJ4/cc-9FCX7S6BP<**:1@:X"cD/XH;GBnOaIt<6+
L51VCN/`jXO,oBarfR/@'9A06NJi[ML4k/.I!KgPBlU?Arn.7%s2t?DqoNqEm5cOU5=7q6:Jt4r
?!q/SBl%^.FE`"XIXm$'L5:\EN/is\OckomPl?sHQN*6MPS01COcYT_MMQt>J:)T_D/Z!+rRq4%
s3(EEotOZmmQ)aX4?u;*9Me\i>@(`LBPVL*FE`"XIXm')Ll%%LNfT<dPa.Q$R$a;0rgNhUs-itU
':>,PQ'@GnNf8mPKRnMqEGh#tr7_1%s3:QHrpZgg475-Y4$>ku8PN&]=Bo6DAnl1%F*;eTIXm')
LPh%NOHG`kQC!u,R[]h<St>bM(7q(eS"#h6Q^*euOH,6TKReDlEfe]Gg])j)c2Rk7Sk8<1.Q]d[
7S6ER<EWX:AS>jsEccPQI=Qs(LPh(OOHYrqR$jD4S=Z@GTq\<Wq4nh%U7n9RSt2C?R@'8'OcPEV
K7A/fK$F)Gs+13qrrVY]9'nYg4$Gu$8l&Ad>$bZLBl%^/G'SLdJq\u;N/j$`Q'[l,S=Q7ETq\<X
V5F3hr2BXns/$O-UnaZXT:VRAR?j)"Nf&UDI!11]q;)(%s3L]HpT7<[3\*9U6:OUD;cd11@V0=k
E--;NI=[$*M2[FUP*D9#S"-(BTq\?ZVPgAlWiGipri-a5W2HMhUS=ERSXZ%5PECi]KRS-!c2-;s
JcF*s!q`Kcm5cd[5=7t8;,p_'?t<qeDK9lFI"-a&Ll7:TPEhK&S"61FUSOcbWN*##Xf\b0pose%
(p:!CX/i4uVkp5bTV%aBQ^!VkLOjc&`VJEkJcF-t!r0!#m5cj]5=7q6:fLP%?t<ncDfU#HI"6j(
M2[LXPEqT*SY)UMUo()iWiW>*YHY79ZEj;4s0D[0(U1*FXK/A#W2?AcTUhR>Q'./`JpWI,qr%L*
s3goLp:*jZmQ*!_5!qh6:fLS&@:a.hE--;NIY!0-MN*^]QC+/4T:r$VVl?_uXfnt6Za7$G[C3O?
\,Ni9[2?9WZE^U9XfSP%Vkp/]SXZ"3OGed@O4F0es+14!rrW,1>jXR+4$H#&9Me_k?"%;YD/jZB
H[gX$Ll7:UPa7c.T:r$VVl?_uY-5+9ZaI6M\%&rY\b`o:\Jr&f\$i]OZa-g=Y,n\'Vkp,[SXPk-
N.uhLh>ZF6JcF7""8pST475-_4$Gu$9Me_k>[V)VD/jZBH[^U$M2[LYQ'e#2T;&-ZWN3,'Yd(OA
[C3QT\[oDbrk/0Brk/6B)S3Jh\$i]OZE^U9XK&7sUnOBLQ]mGbJY26,j8X]1ec,aMe8[Y>3\`][
6:OUE<*<L8ASH"$Fa/7`K8,8COHc'!S=ZIMVPpMsY-G:=[(!QU]">Vg^:q@srkSNLs1nWK*59,"
]Xted\@8lRZa-g<XK&4qUS+0GQ&g`SXQnjGJcF=$"9.G35OCKb4[DP1:K1G%?tF%gEHHGQItNK5
NKBBkR@KnDV5L>oXg#+<[C<ZV]=bej^VI\%rkeNLs2+fP*5K>(^:h1k]"#5X[Bm-AXfSM"UnO?J
Q&piXZhNP\JcF=$!q27<m5cp`5t4IA;cm:4A8#h"Fa/:aK85>EOd)6&SY;gTWN*)(Z*UjI\@T>d
^;%J"_Sa=1rPe`Ts2FuUs2,hn_83q%^:_(h\[])UZa$^9Wi2bgT:2(-Lks.Jjo9o3fDbpHZ:sXH
0K_Th8PW/b>@:uTCiOTBI"6m*MN*daR%'\@Uo(,lY->4=[^`o\]t_A!_Sa=2`l@ttqoJfXs2P)X
s25ko_SO%&^:h.j\[])TZE^R5WMZG_S!T4nQIl)ts+14&rrW/;>jXR+4$Z2)9i>"r?XmbcE-->P
ItNN7NffToS"HCMVlHi%Z*^sK\\#Pi^r""-`Q#s>aSa0Yb5KH^aT':%`l5m7_ns7*^V7Cn]"#5W
Za$^8Wi)YcS<o@pTB?.9s+14'rr`8[E'NbO00;?b85)l\>$b]OCiFH?H[pa(MN3jaR%'_BV5UGs
YHkOE\[oGf^r"".`lH0Bao9KYbl,`bareaHa2c3=`5BI.^q[Rq]=PM\['Hp<Wi2_eS<o=s[e/b_
JcFF'!q;:8m5cpb6:X[F<EWU:Anl7*GC+glL5V(TQ'e)6U8=fhY->7@\@T;c^VRh+`lH3Db0/#Q
rQY>er6>8crlb>a*m)LEa2Z*:_ns7)^:h1j\[SuQZ*(.,V4jHGOH7Z`kl656gA_6NYYORH00MWk
8l&Af?"%;ZDKBuIIY*<3NfoZpS"HFOWN32,ZaREV]Y;1u`5Td=b0.uQc2Q#_ci;;jc2Z#fbQ#^*
aiMQD`Poa4_SEq#]Xt_a[^<<BXJr%jS<o;.hYl^=JcFL)"9$ta475-a4?u>,:/Y.u?t<tgEHZVU
Jq])AOd2<'T;&3^XKJh8\%0/b^qmq-a2lEHbg"GZd*Vd>r6kPks3Uels3;e4bfn8Rb/q`Fa2Q$8
_SO%%]Xtbb[^<<CXJr(jS<f8<nEKt(s4dPTk_24g3\if_77g3O<a&j?BP_[2H$t3sLlIL\R%'_B
VQ$W"ZF%*P]Y;1u`5]m?bKS5Vcd:%ddJ))fd3d5dcd'h\c-4ASb0%iIa2Z'9_nj.']tCqe\$WHE
X/VqhS<T,Sq<J!2s4mVUq5?jP3\NT[6qL-N<E`^=B5DL.G^P$pLl7@ZR%'\AV5^Q"ZF%*P]YD>$
`Q-*CbK\>Yd*_j@rmUkrr6tYnrmCbm,1"Q[c-4DTbK@rJa2Z*:_ns7)]tCtf\$`NFXJhtgRZa9$
lMlG8h>[QR_c,ta0fqTe852u^>$YTLC2\0;H[pd)N/s0hS"?@NWN<;/[CEf]^VRk-aN;WMcHjkb
daQ\EeGRkqeGdqsdf.Yod/VDocHXSWrlZ2%aiMK@`5KO/^qRIn\[SuPYcOh#TUM%Ip?Va0s5*bX
rnMHUmlE3c5!hb5:fUY(@:a.iEHZYWK85>EP*VN-TqnWfYHkOE]"Pkp_oBd?bK\AZdF-IleC=KJ
s47/!rm^nqs3gnos3Uel"4#08bl5d'b/hZD`Pf^3_8*at]=GDXZEL@.US!m=eGAP3JcFU,!pkh1
mlE3c5!qh6:fUY(@V0=kEHQPUJqf/CP*VQ-Tr"]gY-PIF]"Pnr`5]pAbg+P]dF-Lne^aKHrmh%u
rmUkprmCbms3L_j!mSs5rlZ/$aiMNB`5KO/^qRLo]!o)QYH4[uSsH;:m/MY:hu<cSXA&%C0g.fl
92JSi?"%;YCiFK@I"6m,N0'<lS=lXTWiiV6\%98f_8O@8bKS8YdF-Lne^i=KfDjG%ec+(te,Rkp
dJhMmci;;jc2Z#fb8JIEaiDHA`5KO/^qRLo\[SrOYH+RqS<q+imJhb;i;WlWdTZ'"1d"&m8kr;d
>[V)UCiFK?H[g^(N/s0iS"HIQWi`P5\%95e_SjL:bK\>ZdF-Opf%/I)f\$2T!7q/"!7_"srmLen
rQtPirm(Pgs31Jcs2tA_'un53`59@+^:_%f\$WHDX/DYZU$)FCs+14.rrUu#4RY<c4?u>-:/Y.u
?t<qeE-$5MIY*?4O-5fuSti0^XK]%>]"Gep`Q-*Dc-Obbe'uq!f@U#Ss478%rmh%u!7UqqrmC_l
rQkGfrltDcs3(Gb(Wjb=`l>s8_SO%%]Xt_`[BZm7VOs9kp?qs3s5EtZpm=S?3\WW[6V'pK<E`^=
B5;F,G'\RfKo(ePQ'n29V5UGuZF.6T^;7b-aihrTdF$Fme^rFOfEp0\f%&:!e'c[lrm:YjrQb;b
q98i])9U(CaN)<>`5KO/^V.:k\[JiLXf8(dUuMCBs+140rrW&-8FJSm4$H&'9i+ko?=IM^Df^)J
IY!3/N0'9kS=lXUX08h:\@]Jk`5]sCcHstee(*%$rR_P1f%/C$e'c[ld*U+brm(GdprrWY!6kGc
(s9tBa2c3=`5BI.^V7@m\[JlLXJhn`\b,@jJcFd1!rgbamQ)p_6:X[F<*3F7Anc.'G'SLeKSYPI
P*_W/USb&nZ*h*R]thP*aihrTdF-Opf)4//f%&:"e'cXjd*L"_c2PrQbQ#fdb8ACDaN)?@`Pf[2
^q[Rq]"#/RYH+OmUYGb:s+141rrVYF5OUWb4[DP1:K(>"@:X(gEHHJSJ:r]8NfoZqSY;mYXKSt>
]"Pkq`Q63HcdC1if%0iQs47_2e^N!qd*L"^bfn8Qb5TNXaSEsUarJULc-4AQaN2EA`l5m6_83n"
]=PMZZE:.&S[%hVs+142rrVr":%(+p4@)D.:/Y.t?t<qeE-->PJ:iW8NfoZpS=l[VX08h;\\,_p
`Q63GcdC1je^j`P&_GU`e'cUhcHXPTaiMQDa2Z-o`VmgSa8O*ZaW8OJcd:"_b/hWD`l5m7_nj.'
]Xtb`['?a3Tq'a5nc+1?jSo;0?p!j=/NZ0b8PN)`>@1lQCN"9<H[g[&MN*^^Q^XP@VQ-c'[(*`_
_SjL;bKeJ_eCE.LfFZWaeC)^ic-+5L`l5m6_ns:h_>V7L_u@UQ`W!mV`u`=Dbg=efda-(Wa2Z*;
`5BI.^qRIn\@/`JX/2Aro^Vs4s5j7^l[q@i3\<EX6V1!L<`ra<B5;C*G'\UhKo(bMPF%`/U8=fi
YdCmN]thS,air&WdaQatrn%_3e^MpkbfRoD_nj+%]tO?VrOr6Fs1eTLs2"]OrkncSrl#u!a3)]W
g"P')bK.]C`Pod5_SO"$]XkV\ZEC1%Vr%FDs+144rrVqo9(4l24$H#&9Me_l?".D\DK9oGI=[*.
MiX'fR[p+IVlR#,[^j&c_T'^@cHstfe^iC*f@JI!ccjJK_7mRn\@8rU[^ENO[^WaF\-fdX]=bei
]tV7r_#D1M_Z.OR`#cn=bgbG,i7cW%a2Z*;`5BL0^q[Rp]!o&OXf%o0q==Q:s6'C`r5Mb*3]oSm
7nc`Y=^>EHBl.j4H$k-pLPh+TQ'n/7U8FrmZ*_$Q^VRk0b0A8[e(*%$f[n[&d*0PH]=5,NY,n`(
X8f7KXfek3Yd1R@['d?O\@K2_]=bhk^;%J"_8=(,_o9aEgZ.GWhUK`hrl#Ge_ns7*^V.7i[^30<
Uo3GUoDaCAkPkVGI6I'\IQ^"e92JPg>[V)UCiFK@I"-d'MN*a_R$sV@V5UK!ZaRHY^r+16bg+S`
eCN7(f%&3oahtm&Z)41aS"#t@T:r!SV5C/hWiW>*YctF>['d?O\@K2_]Y2"m^V@S#_8=49f]25^
kMOb1`l5p8`5BI.^V7@l\@&WFVkVQ8oDaCAkl1_WPXFq$Im$(e8l&Af>@1lQCN"9<H[gX%M2dU\
Q^XJ=Uo18rZ*h-T^V[t2bK\D^e(*(%f$r*md+74Am-<BJVMg%\MN3g`Q^O>7TVA6ZW2Zi!Xfnt7
ZaI6N\@K2_]Y2%o^qde'a3if/nF5c%d`BDH`5KR1_8*du]=PJYYcFV-n+6Rcrs/Q%qXs@BjP0Xd
$/FS'lL4B>qY^?;rrVo'9COu#4@)D.:/b1t?Xm_aDfU#IIY!0/N0'6hR@KqFVl?f&Za[N[_8OC9
bg4\ceCN7&e'ZV'nbW7as#'fRn'.>HF*iImMN3gaQ^XJ;U84ZaWN3,'YHbC>['mHR]">Se]tV7s
_ogWonFQ59h9X-\rko;a_SO%%]t:e_ZEC1(fDbFB\,Qj8r:8pr]qg[;DeE>R=q"\LC3"`\Q_q:B
l1Y&&rrW2EAFMW:4?l5*:/Y+r?XdY_DK9lFI=R!+MiNscR%0eCV5^N!ZF.6U^r"+5bKeJ_eCN7&
eC!%<rV6CJrU7IRAo2^=JVK&BOd29%SY)XPVPgDoXKJe4Za@-K\@K2_]Y2%p`6[<.oCVS;g<@P8
`;daT_$[uk]XkSZZ)jb3p[nKurt#,+oARflO)\B!7mT=$4$3=#s#^6C$pHmA78[K=Q*ep#p&3MB
"9.M/54LZj4?u;+9i>"q?=IJ\D/aWBI"6j)MN3d_R%'\@Uo:>rZ*UsP]u%b0b0A;]e(*%#eCNgY
rq6:Yqr<iI?ts_,IY<K8O->ltS"?:IV5L8lXKA\2ZF%$J\@K2_]Y;2%eE?App$q4sai23:_o'@-
^q[Rp\[JiJW2\eTp&Dc+#lXAI\rg]%5s+6j#<YCSDk5&;q:YcLo:SMA3\rl`6V'pK<a&g=AS>q"
FEi.^K8,5AOHZ!!SY;jWWiWD1[^j&c_oBgBcdC1jf%/@'m/6DR0Bo$/<aoibH%(C$MiNsdR@Be@
U8=`cWiW>+Yd1UC[^`lZ]=bu&hX0t/p$h"ga8X!e_ns:,^qRIm\@&TCVTJW+s2"^?rqG6LHW<Kt
l8nr,pcJdH:2>p'h=CL#rrW#%:[pJ%4?u;*9i4nn?"%;XCiFK?H[^R#Ll7:VQ'n/7U8FokY-G@C
]=u,#aNMiTdaZgteCsEjnGb3AnXicU>%DGiH%(@"MN3gbR%'Y>U84ZbWiN8*Z*L^D\%'#\]u&4Y
o(MhNmG,q(rPSZP$GaEj]XkSYY,Sbnp&Dr0#5mu;Q=EBHloP/.#Wt775X@_(6U^lA$q*EJ5XS7T
K!<epq>K@R"8g,H47YEg4?u;+9i4nn?"%;XCi=B=H@CEuLPh+SQ'[u3TqnWeXg,4@\\5era3)WO
dF6UqeCa6gmJeg+SkD[U?taJ$H@CL&N/s0gR@Bb?U84ZbX/rG-Z*UgG\%0)`a4]hPp\4:Be].R*
s2#/[^q[Rp\@&TCWRq.Cs2G!Ap>E)u9K_Z&%6H[95sdt/7nH<G9heC\:C^,o91h]C6:=[iOh'<?
rUp0qmD><GKpomGqu6crc=/m+3\rl`77^0O<a&g<ASGt"F*DqYJV8f:NffQmR[p+HVQ-`$ZaI?U
^;@k1bKeJ_e()t!mJZAN0(P?$78d?#CN4NFJ:rc<OHYutS"?:JUo12lXKJe5ZaI6O\\H>Gn+QPM
n);L/rPSZP$GaEj]XkSZYGf22pA`56#6+DPS8Ck]m5bkA4?c"s6q9gA9MSD];H-Uq<`]0#&6;e1
;c-=`84H'FC6?$Xo_eOb$/()#)?pBa6AO7mrr`2IE]rtS1H[oj8PN)_=^>HIBP_U.G'SIbK8,8C
OHYrtS"HFOW2m)+[(!WZ^Ve(6bg4\beC;q-qs+26roAl(78I)qBlA*>IY3E5NffQmR[p(FUo()j
XKJe4Za@0N]#)nXoCr"NlI<lQ_uIR\_8*dt]=G>SWj(j`s2k9DpXY^954(BF4$5Yk6UaO<9MSD^
<*!('>?b?<?M\"@?<pc==B8?r8OZ!:<.qU5o)/Id!nhW5qZ$m?F4pTnqU9.Bo/\]i4[DP0:Jt4s
?"%;YCiFE<H$t0oL5CkMPEhN*StVsXX0&V4[^j&b_8XL=c-XnfeC*"CkPm-nH9OVg<F06QEcuh\
KSYPHP*ME(SY2^QVl?\tY->4=[C<cdfB`)'p@7%c`;RR^_SO((^:_%d['$I_rV-?8rri2UU0g4Q
3YjhA5X\(69M\Ma<EE=-?!^oI@Uiq[B)6-UAn5C[?s[&><)?4Y7nmKJZe4[@m"ZMT!!X?sm.J)-
4nC]l4$Q)&92AJf>@(cMBP_U.G'SIbJqSu=O-5cqR[p+HVQ$Z#ZF%-Q]thP*air&Xe'ujto`"%R
/cF(#4$lS>?"@\gG'\XjLPh+SPa@f.T:r'XW2cr%Yd(OA\%TrAn+HDHkgmfO_@",p^V.7i[BZgP
q=jp5rr_i+E(91T(d'uI6q9jC:f11m=^,0>@:EbZB5)*qCi'&es).jR'PqQ"BP(gb?X$W3;,0_S
:Ln`rAL[?R"@8u[=$6(82*F2m85)iY='K!?ASGt!EclYTItEB0MiEg^QC=>9U8=fgY-G=A\\#Sl
`5p*GcdC4jdc^Vks"4&C8io@4<*a$KE--ASJqf,AOHYuuS=ZFLV5L>oXg#(:[CF$"k486=mbc7+
qn`ZU^V.:k[^3*En+Zk-rri#CIRuWg3ZgIJ6:FF<:/Oqj=Bf*>@q91bBkqR'DfKf=FEDYJG'<h,
&9Dl3E,KK,B4PFW>?Fm&9fVbF!Y71So/\ck5=7n5:/Y(o>[V&SC2Ip2G'SIbJq]#<Nf]HjR@KkC
Uo15oYd1^I]=u(u`lZELd*gCjh"g"1/F8H]4$cJ;>@D2^F*E"^KSPJHOd)6%SY)XPVQ$VuY->4=
\AQq`o_%e<e].L'$GjNm]t:h^YcZXXq#A_@"TIklA3TBA*]lMM6Us^B:K(:r>?tWHAS5^mD/O<5
F*2VLG^4U]qKid"H$FOUF)c,8C1^p8,[=JJoDemm*qVKP4?c&!85)iY='Am;@qTOnE-$2JI"$[#
LPh%OP*ME(SY2aSW2m)+['mQY^V[q0b082ZdaHe=ir:Y0ePnRl6;(6\@V9P!GC+jnLPq1TQ'[r0
T;&-YWN*&'Z*LdLakuRZoBb5R_SQ/e$,OEk]XkPWX2t)2s3^iKqU9m^loH7N5=.e19M\Sf=^59A
@qKChD/XE7F`r"TH@($gIt3'#r.=s"s*tZ1I=$9bG&q'u!!#u/@2faYpAb3p*qq]V4$5_o77^'I
;H?q*?XdS[CMn-6G'\ObJqSo:NK96eQ^XG:U8=cfXg#.>\@]Jj_oBd@c-Xkde*6his"=2@6TR=r
8lAi#B5DR2H[gX%M2dX]QC488TqnQaWiWA-ZF7HljR;[-im,HX_>V4W_8*h!]=>5N\*`tje,KRM
o!N'om5cFQ4[;D*92AGc=Bf*?A7fOkD/aN;GBeF\I=?]tJqJ`1L5,S\re:u<K7e`)I=$&T!!"Wg
CM,a9pAb:",<J!d3r_9(4?Yko6UjX@:Jt4q>[CiLAnc("EclSPI=Hj&LPh(PP*D9#S=cOOVl?c$
ZF%*O]YD>%a3)WOd*gA"qVV<'h.o)54@;b?>%)#YE-6JUJq]&@OHYuuS"?=LVPpMrY->4@_Udl=
n)i-=rk\TNs1nuU^:_"bZ*E6dq>\tE"9.GP9'eS`4?c)$8PN&]='Am<A7fOlDK0`?GBnRaIt<6*
KnbA=M2I4Lq2>oDMMd4FKmJ?$EYD&hG&qY&)ts>#&i;sB6:!n(6UX@58k_uV;c[")?"%5SBPVL)
F*2_RI"-a$LPh%OP*D9#S"64IVPpMtZ*UmL]=u,!`Q63HcdC1mnD=0qqQi^R3^#l);d*UACN+EA
IY*<2N0'6hR@BeAUSXohXKJh7\]EIgmHE6ErPAKM$,OEk]t1\ZYLEIGs4.,OqqZfoloHCS5XS%7
:/Y+q>[LrOBPVL*F*;eSI"$WuK8,/;MMmFQNfK0^q2l;OO,f6[3Ym*tKRnZ&I!^*T4U_#b";FI(
:/1[\4&&IH;H-Xt>$PBB@qB=gD/XH:GBnOaJ:`K1MN!RXPa7`,SY2^RVlHi%ZF%*P]YD>%a2uQN
d*^;$qVD0)kB/pK3^-#-<*NdCCiOTCIY*91N/s0gR%0bAUSauiXKSn9^!#<skM4:u^qde&rk\rV
^V7@k[BIIEq>]%G"8pJb54(BZ4$>kt8PDr[='Ap=A7o[qE-$2IH[UBpK8#):MN!RVOHG]iPa.N"
QMm0MQN*3^P`nW<!#L<HLkg\:Jq&2^2@T?\";Fs@=BGH%3`fHf>?tQC@Us(aC2@d-F*;eSI"$X!
KntVFO-,WlR%'V<U84]dX0/_6[^j#`^r+.4b0A5YdF@[bh>[QR[nH]],sOjo<EimECN+EAIY*91
MiX'eR%'\@Uo(,lXKJk<^X:s"hTs+=_#M7N_?dun^:_%cZ*a,us4@8Qotk6&m5cRU5!h_2:/P"o
>[LuQBl%a/Fa/7]ItE?/Ll.+MO-#NhQ'Rc(R@4&Cr1!_Ts-XLe>6+^UO-,H^Mi!7EKS"#-+:S2K
!!E3#";,!M?srt@s'Rco@Uit^BPD7!DfKi@GC"UaJ:`H/Ll.1QOcu&tR[g"EUo()jXfo(=\%98f
_8O@8bK\A\d+eHPrrM>(r]2r!5Y+[N>[h>^EHQSVJq]&?O->ltS=ZFMVQ$VtYHtaWe(rd+`PB7(
_84"*_83n"]=>2Mir8fSf`)'HZX,c]3[HjO6:OUC;H?q+?t<nbDK0`@H$k*mKSG;@N/j!^PE_B"
R@9V8rgj(\r1Et['V1baC,2'@Q'7AlNf8mRLkg=d9e?8D!?F#r!!<aCBP79U2eQf<BkqO%DK'W;
Fa&+XI=?a"KnkMCNK93cQ'[r0T:hsUW2cr&Z*^sM]=l"t`Q--FcHstbkM#q;qkcUW3]0#d8l/Sp
@VBUuG'\UhL5CkNPEqW,T;&-ZWN3/+[(OK0f?hOQ^V@S#_8=%)^qRIl[BIsSqZ#:L"9._S7dN/a
4?l2'92JPg>@(cMBl%a0G'J@_JV8c7MiEd[Pa.Q%R@B_;StMdNUAgqcV#R2)O)^keT:_[ES!fV.
P`h/jNfAsRLOK.%![2n^rW!'7DK'N4rbhaS1iRGTF*)SLG^=^bIXm$'L5:_GNf]EfQ'[r0StMgS
VQ$W!Z*UjJ]"Gep`5]pBbg4Y_g@s;#!T%2^3[?jU92Sbr@q]_"G'\XiL5CnOPF%`.T;&0[WN32+
[_9c.cH!d#^C7oo_SX.)^V.4fZ+or-s4[JSn>8nG3\3?W6UsgH<**:2@qTOnEHHDNI=Hm'Ll..P
Ocu&sR@B_;T:hpRUnjiaVPg?iW#hk3+9>B7USFQVT:MI?R$O#%PECocN/@c=!,-q5;>pOu*d2^L
FoHL_FYe2bGBeF\I!pKqJqSi5M2R@ROHPinR$sM9T;&-YW2cu'ZF%'N]=u(t`Q-*Ec-Ob`k5F!5
!qVI;r]2kt5=S=F=^GZRDKC#KItNK5NffQmR[p+HV5L>pYHk[RaNDN=^:q@t_84"*_8*ds\[8p6
qZ#=M!qr<_loHUY5=7t8:fLP%?t<ncDK9lEH[^O!LP^tNOd),uR@Kh>TVA3WV5C2jWN,lss/c7$
*`E%=!g'@YVPKu]TV%dER@0D-Q'7>k)?9bTLP:CWrVusSI/\KlHN/?lHnTnsIt<3(KS>2<Mi<XV
Ocu#qR$sM9TV80XW2cr&Z*UjJ]"Geo`5]m@bg+S^eF_Dn!r^2:rAl/b5Y"RL>$toVDKC&LItNN7
NffToR\$1JVPpPsYdD!X`P]Lj^'qcl_8=(*^q[On[^=rgs4mVVn>])Bm5cd\5X\.::fUV'?t<qe
DfU#HI=Qs(M2R@TPEhN(S=Q=HU84W`W2ZetXKAY.q6(m10*#I_X/`.tW2?DdTq@pIS!ob4QBWKl
!,RUOM2+(i!!-#Lq1(/MJqAW.KnbD>MN!OTOHPfmQ^F53SY)XOV5L>oXg#+<[^j#`^r"%0aihrS
d*U\JfDbp<F?KL_,!A:c:fU_.Anl7+G^P$pLl7=WQC+26TqnQbX0/_9]>27trk/BI^qp#e#Jn0g
]=5)nrVQWNrrVhg;!^4p4?l5)9i4nn?"%;YD/aQ?H@CEuLPh%OP*MB%S"64GUSXidWiN5'Y-5(7
Z*O54s0<rUA$N8AY-"e+Wi;nmUnXQUT:ML@R1'&q2Na&[N/An]!!-hgL&H]'KeA+8Ll%"JN/`jZ
Ocu&rQ^O>5StDaPUo12mXfo"9[C<`[^;7_+aND`Pcd:,*rn%/Oqk$4Q3\iog92\hs@V9LsFEr:c
Ko(bMPEqW,T;&-ZWiWD0\%BAf]Y(ql^VI\&_SO%%]Xb>dqYU<Lrr_qs:-7i(/Nc6b852r]=^>HI
Bl.g2G^FmkKntYIOcu*!S"61FUSXlfWiW>*Yd(L?Za@+9[4$'t[C*BKZa-j@YHG"/Wi;nmUnaZX
T:VSV!!"X:P`h/iMA#si$\+$0Lb=OAMMmFQNfT<bPEV8tR$jG6SY2[OUo()jXKJe5[(!TX]thJ&
`lZEKcHjhip=90Db>nI$+?Vq\:/kA'A8#h"G'\XjL5M"RPa@i0TVJ?^X0&V5\@_^P%DBNl_84"*
^q[Ro[(H#.s53hXqRBPl3\NQZ6q9pI<EWU9ASQ(%Fa/:`Jq]&?O-,ZnR[p(EUSXlfX0&P/Z*L^C
[C3QR\@DII".''"ErNNl[^EKKZEga=Y,n\'W2HMhUS=Fa!!"dBR$X)&PCE]8!$mNuNW5%=O=66S
PE_>uQ^O;3SXuIIUSOccWiN8+ZF%$L]"Gbm_SsR;bK\>Zcf4lQrrMJ6q`5i_6V:3U>@;#WDKC#L
J:r]9NfoZpS"?=LVQ$W!Z*h*Rrk&]R^;%M#_SX.(]t:_\mJHqArrW#3=R8(%4$Q)&9Me\k?"%>Z
DK0fEI"6j(M2dU\QC+/5TqeH_WN3,(Yd1UB[^WcW\[oAar4FAg.f]Q+]=PSa\[T&U[C!9GYck.0
Wi;qpVPOe'!-=a%S!o_2Q=g*r!D8OOOoLUEPQ-jpQ'Rc(R@9V8SY)RLUSOccWiN8*Z*UgG\\#Pi
_8F:6b0/&Ud*M(Xec,^JY<DeE1-7]h9N"qs@:j:oFEi4bKnt\KPEqZ-T;&0\WiWD0[^`o\]=bkm
^qmn)_8*dt\?Y#lr;YaU"9-nf4mY3^5!h_3:K1G$?t<tfEHHGQJ:iT5NKB?iR@BeAUSauiXKJb4
Za@0M\[oAa]Y(rV^AkqO^47"$TtUH<,.kFs\@8oS['R$@Y,n_*WMl^+!!&+ST:VUCR[KO/!!!%E
pltfDs->!uR$jD4S=Q7DTVA3WVPgDoXKJe5ZaRBT]Y;1u`5]m@bg+P]ch#Zd!rgV<q`6)e6:ajN
>$b]PDK9oHIY3B4NKKHmR[p.JVPpPuYdCjL]">Vg^;.Sa_?Rik]t(E!rVZ]Trr`8WFZf"O00MWl
92JSi>[_2XD/j`EI=['-MiNscR%'Y>USXohXKJe5ZaI9P]">Vg^;%Fu_"ker]nF2:^V@Lr]tCqf
\[],W[Bm0CYHG"/WrrD&RAH[QTq@pIS=1As!#(RlQN3EPR3nAlS"6+ATV/!RUnsrdWN3,'Yd1UC
\%0,a^VRh,a2uNLcHsneqU>HFmq)>V0Kqls:fU\,A86"'GC+jmLPq1UQ'e&4TqnQcXKSq;\%0)^
]tV7t_8=(*^qRCdb52&9ir9)LN'ckl0K_Wi8l&Ae>[V,WCiOTCI"6m*MiNpbR%'\@USauiXg#(:
[C3TV]=bkm^qmn)_o)Dj"/NlQ_u@OP_%FJt^:h1k]=GG\[^<BHZ*:F!rW!F'W2?DeU8"<QSq[=I
!(HUNs-tC+StD[KU8"EZV5L5jX/rG,Z*LaF\@T>d^VRk.aND]Ncd9tmrR:cJq1_*13\WZ^7nll_
?".G^E-6GSJVAr?OHc'"SY2aSW2m&*ZaRBS]=bhl^qmn)_8*ds[DM\9s5X+\p9$pP3\``^6qL-O
='K'CBP_[1G^P$pLPq4VQ'n/6TqnTcXKJe5['mKU]=kqn^qmq+`5KX6rPfhj&MK<d`5KR2_SX.*
^q[Us]Xtbc\@8oR['HrWrW!I"X/W%pV59u^TqGTi!!.ZMSH,2XSc>;]T-U)'Tq\<XV5C/hWN*&%
Y->1<[C<]Y]Y;1u`5]m@bg"J\dHUYZrrW2#7f#.m5!qn<<E`a?Bl7s8H[g[&MN3jbR%'\@Uo15o
YHkRF\@T>d^VI\b_?Icj]XYWFr;YjX!r]i;loHg^5=A(;;H@"/@q][uFEi1aKSYPIPEqW,TVA9]
X0&S1['mKU]Y2(q_8=+.`Q#s=qoBao!!!hQ`l?!:`5KR2_83q%^V.:l]",>[[^EJ;rW!L$XfSS'
WMcYjV50k/!!"%2qk*h[-)%'-USFW]VP^;jWiN5'Y->1;[C3TV]=ktq_o9[<bKS5WcdVI4rrL5A
qDorb5XnFE='T0GCN"<>I"?s,N0'6hR[g%GVQ$W!Z*UmK\\#Mg^VTlc"Mh^]\a97ojSo;ECd%,G
/3ZBj9Mnem?=IP_E-->PJ:r]9O-5fsS=cRQW2cu(ZF.0P]=bkm_8=+.`lJ)"!6P,\0Q@0pGfe%J
a2c3=`Pf[3_SO%&^:h1k]",>[[SHl'/[2I^XfSS'WMl_lV2>9R!+>f#!29=bs.fUis/#am*iGs9
WiN5'Y->1;['dBR]=bkn_8F75b0/#Scd(/:dJj:)BfZ&O-TaL[8P`>h?Y!heEcubYJqf/CP*ME(
T;&0\Wi`J2[C<]Y]Y;.s_>_:S^qI=hkktJBrrVYR53t<_4[MV2:fLS'@V0CoFEi1`K8>GHP*VQ,
TVJB`XKJh7[^`o]^;.V'`5Ta;aND["b:pG)a32WIaiMNC`l5p8`5BI.^q[Us]Xtee\O6DtBsY+M
Yck12X/i8"W2E>i!!\bbV#R:jVZ3P9W2ZetXK8P,Yd(L@[C<ZW]=ktq_SjI8b0/&Tcct>EdJj79
L&!b^4$Z8.;-$q1Anl7*GC4poLl@CYQC=A;USb#lY-G@B\\#Je^VI\&_ns4']=6bss5s=_qQWi_
3\WZ]77p?S=Bo6FBl7s8H[g['MiNscR@U"GVQ$Z#ZF%*O]=bnp_o0O7aN;TJbKS3+c2GlcbQ#^5
aiMQD`l?!9`5BI.^V@Iq]Xta"!!%kM['R'CYck43XK/D%UC.&%2>O-KVuN^qW?8.@X/rD)Y->1:
ZaI6N\[oGe^qmn,`lQ<Gc-FY^lMf32!r/BMq)U,l6V(!O=^GWPCiOTDIY*<3NffToS=cOPW2m&)
ZaI9R]=kqo_8=+,_8!XlgA_'NkPkV^cs?!!00;Bd8PN)`>@1oTD/scFIY*<3NffTpS=lXSWN<;/
[(!WY^;.S&`Q-'BbKJ,Sc2Q#fci;8pQY=%Vc-4E1b7i%?aiMNB`l5m7_ns7*^V7FpYlXh(%LBMN
['R'CYck43XK8IMrVutMWVi^mWW0!tWuS4CY-5(7ZEppH\%'#^]tV:u_o9[<bKJ/UcIMR6rrW,.
62<Po4$Z8/;-$n0ASQ.)G^P$pLl@CYQC=A<Uo15oYHkOE\\#Mg^qmn*_ns4&\CLTWs6'C`n;0Es
3\NW_7nc`Z>$kcPCiFNAI=[*/NKKHmS=cRRWN<;/[C<`\^VRe*`lQ9Ebg"DXcd2F6#rqtdNm>qp
c-4E2b7Mh<aN2EA`l5m6_ns7*^V=de!#,UH\$i]OZa-m@YHP+2X>U%Y'WA)2Wr&jrX<"FGY-5+8
ZEppG[^`l[]Y;.s_SjF6aihoQcHb/>d/O1Hcq='o/NuNn:/kA'@q]_!G'\XiL5M"SQ'n/7USXrk
Y-G@B\@T>d^VI_(_ns4'\^13Ps60Iar3o2_3\<HZ7S6KV=^GQLCN"<>I"6m,N0'<lS=lURWN<;0
[CNo_^r"%/a2uKIbg+M[rm:Mh*jYkI!4C.ucHXVXbfe2Pb/qcHa2c3=`5KR1_83oKr;[K9]",>[
[^EKKZE^X<YHFDu!!$L)X8K%@Xfeh1Yd(I>Za@-K\%'#]]tV:u_o0R8aihoQcHYDJci4%,>51-1
4@)G1;-$q1ASZ4+G^P$qLl@FZR%'\AUo:;qYd:aI]"G_k^r"",_SO!ubPV8<l2LhahJV[@/iu9c
8PW2c>[_5ZDKBuJItNN7O->s#T;/9_XKSq;\@]Gi_SjI8aihoQcd1"dpsKSub_\,>d*L%`cHXVX
rQ?S2aiVWEa2Z*;`5BL0_)_Pr7_HCG]=PP_\$i]OZa-mAYcq=3!!o:oXo>L'YQ(gBZ*L^B['d?O
\@K5a]t_A"_o9X:b0/$.blR21ci4%8D>6.<4$H&(:K1M(@q]_"G'e^kLPq4VQC=A<USk,oYHkRF
\\,Vj^r$)g"Mqjb`V9?2l2LhXR76I'/j2Nk9Mebn?Y!heEHZYWK8,;EP*VT/U8=ijYHkRG]=u(u
`5g!Bbg"GZd*^:iq:#Alrm;%pd*L%`cHXVXbl5fcb5]R.aN)?@`Pod5_ncla!1gj;]Y(kf\[f2X
[^EKKZa-j;%fcT(q6(sGYHP17Z*L[AZa@-K\%&u[]=kqn_8=.0`lQ<Gbg"AYoZmI<pk1a(3[?gS
7nll`>[h>]E-6GSJVAr?Od2<(T;/9_XKSq:\%0,`^;.Vb_Z[il]YX4<s6BUcr5DM!3\3BZ77p?S
=C#BKCN+B@I=d01NffWqStW!ZX08h:\@]Jj_o9^>bKS8Xd*^;@e,7_pdfe+EdF$>?ci;;jc2Z#d
bPoZaaWnmIa2Z*;`5J,F!"Zl[Ye7Z^]=PSa\@8oS[C!9HZ??3J".].js0;U0s0Ma4(U^]\\@K2_
]Y2%p_8F41a2lEHbg">]qTf*Br1H:K3[6jX92JVm?Y*qhF*Dt\KSYSJPa@l2Tr"]gY-G@B\@]Df
^r$)g"N%sf_!q=&lMgqQKKSW_/iu9c852u_>@;#WDf^,MJ;&i>Od2?)TqnWfYHkRG]=u(u`Q63F
c-F\`daQ[prRLns!7Uqq$;i?o^!G!Lc-=JVrQG)\s2bAaa2Z-u`?-i8!';;'!3NuK]tCth]",>[
[^NTN['C\P!'miqZ*F54rj2U2s0WZO\%&uZ]=Y_h^VRe)`5Td=b0/#Qb1QX;rrW2+6h`Vl4[Ve:
<*<L:B5DR2H@CI#MN3mdR[p+IVQ-c&ZaRBT]Y;1t_o'F1_7mbSrVu3`!rfMtloH[\5t4LC<E`a>
BPha4H@LR&N0'9kS"HIQX0/_8\@]Jk_oBd?bg"J\daHUoqpk_rs3q"r$-`nZ!2@c`c-4E0bPoZ\
aX,$K`l?!:#QOj(*WQ25^V@Lr]Xtee\[f2Y[^NTNKE(uQR^9?+qQp4/s0_m8s0i`R]">Se]tV:u
_Sa@4a2uNJbfn8grm(NFrkhA&3\3BZ7SHZ\>[V,YDfg5PJVAr>Od)6'T;/9_XKSt<\%95d^VRh*
`5BF+_<h1#s6Tafrp+D_m5cd[5=A(<;cm:6Anu=,H$t:!MN3meS"HFPWNNM5\@TAh_oBd?bg+P^
daQ^rr728+eC2mpdF$=>!!!i7bl,`ab5BB\aT'B^aooi-a2Y4[!#_gg!"PL6^q[Us]tCth]",>\
\$i`Q-NF-)pU'q-s0``Q\@K/]]=bei^;.S%`5Ta;aihlObKK(mrrLJDpGsie5"&"><EW[=BPhd5
I"-d)Mia-hS"?@NWN<8.[(!WZ]t_D$_o0I0^VT47rr;?b!r0#eloHX[6:X^H<EijBBl8!9I"6p-
Nfo]sSt`'\XK]%>]"Pns`lQ?JcHstfeC<%HedU*Ye'cXkd*L"^`.377bQ#`aan<^pb0/#PaiAB$
!#`_b_o'@._8*h#^:h1l]=PSa\Gir;Q2Ua`P5)Ha[f<fK\@K/]]=Y_h^;%M$_o0O6aN=A'!:%f.
!SUZL3\*?\85<)c?".G^E-6JUJqo5EP*VQ-U84`gY-G@C\\#Pi_8F1/`59=)k5PA[m/I+FFmM.>
4?u>-:K1G&@V9LsFaALhLl7@ZR%0hFW3!2/[^j&c_SsU>bg+S_e'uq!rn%2$s3qA'dEp4ccHXSV
bPoZaaoBH\a7mR]a3)WOccjN]!!%2]rko)[_SX.)^q[Usrk/9C#eIRW\@=9o!%[iQrjVp;s1&ZO
]=Y_g]t_A!_Sa=3`lQ:$b6#i1m`b\3hd6";.m$$e9Mnhp?tO.lFEi1aL5CqRQ'n2:Uo15pZ*^sM
]=ktq_o'F2_SEqIrr2rbrrW+s8*i8d5!qk8;HI+3ASQ.*H$t6uMN3meS"HIRX08h:\\,\o`Q63G
cI((heCN4Mf+?N`eC2glcd'eZbK@N;aN)@"`r*gM`#H\9bL"_eQ2gme\AcG*['@Nc_SO((^V7Cp
]Y+6U"1l%RS,NBfOSQBb\Gj#M]">Se]Y2%o^qmq+`5]j=ao9Hbaj0M%rrM:rp,X]c5=J1?<*EU=
Bl7s7H[g^)NKKHmS=lXTWi`M3[^`r_^VRh*`5KR0^uPFns6]gfk%6^B3^5_l78$HW=^PZPD/sfH
J:r`<Od2?+U8=ikZ*_$Q^;@h0b08/XdF6Urf@\^,e^W'rd*Kt\bK7lIa*I_0HGoa!pqlsH*5oh;
cICGo!!"an`dIge:;aud_SO((^V7Fq]Y+6Us1/?D5l^m5ppg=6s186Bs1AuX^:qD!_Sa=1`lH0B
b0.rLf([Dg!U+:k3\!9[8PW2d?".G_EHZYXK8,>HPF.i3U8FokYd:aI]"Geo_o'F3_nj.Krr2rc
rrW&&6gQiu4[Vb8;HI+3Anl:,H%(@"MiX'hS=l[VXKSt>]"Ytu`lZHMd*gCnf%8R,f@AC!dEg+^
b/q]D`PfYb!!$ZJo>(=@s2#eo`QQ]W!<<+8`l<`5!'A-:_ns:,_8*h#^:q:n]`#GFT`G,nNqp6b
])K>A]`5YW^;%J"_Sa=2`lH0Bb0.rKh>>Fr!UY"%3\!6X85)o_>[h;\Dfp>SJqf2EPF%c1U8Fok
YHkRG]"Geo_o'I4`50:Prr2rcrrLtiloI?o5t4LD<E`a@Bl8!9I"6p-Nfo]tT;/<bYHkXK]thP+
air&WdaZguf@\^+eC2gkc-",J`5BI-^V7EL!!&/V]`,VD^AktI_#M4]_Ss`E!!!)CaMsYK%7\:S
rkf,\_83q&^V@Lr]tD#W])t4s!%[iUrO`'A'>;,p^VRb'_o0O6a2lBFb/q`arQP9BnU'pi.Q]j`
92JYn?tF(jF*N(`KSkeQQC=D=V5UGuZF.3R]YD;"_o9U5_SYI7rr;He!rAugm5d6h5=7t9;cd76
Ao)F/H%(@#Mia0jSY;mZXKf.B]>)8&ai_lSdF6Xsf@\^+eC)^hbK7`@^qRLo])K5C\@%C(\,Ei9
\c98?]DoPD^&GbG^CS,s`>ejX>gU[8_o0O5`5MYms2+fPs1ecP^:q:ork&9DTDefjQhJ&j]G/&n
^;%J"_8F1/`Q#s>ai_cJa5m3CrrMY:oK"Eb6qL0S>$kfSDK9rKJV8l>Od2?*TV\TfY-PFD]"Geo
_Sa@3`59FSrr2rdrrV8!47,'g4[DP1:f^e-A8,q&G^P'rMN3meS"QRUX08n>]=u,#aNMiSdF6Xs
f@\^+eBuRba2>[)\[JlOZMCd*ZN%91ZlHK^[^W`U\[f;`]=bei]tV:*!!!8Lf$MX_a2Q$s`;[[S
_Z.IP_#M1L^AknG]E:4q!&jVarOi0D'>D5s^qmn*`5KX7a2lBFb/hQ_rQP9BnU'mh.6U$h:/b8$
@q][uG'ealLl@F\R%9nGVlQu*[(!WZ^VRe)`Pod5_W:^qs7$$ir3o>d3]K2c6qL-P=C#BKCiOWF
ItWW:Od2B,USb&oZF7<W^r4=:bg4\ce^rF*f@A?rbf7K2[^!!9WMlcpV^eX4RAm3dX/rG+Y-5(7
Z*L^C[C3NQ\@K2^]=Y_b"98F>h;$T0cH<GQ\GjGG_uIUQ_#hEe^](tI]`Gd6rVuqmrk&6Cqn2pA
s1SlU^qmk(_o0L4`lH1"aTKT$hYk[u!V(O/3[m0W7nll_?"%>\E--ATJqo8FPaIu5Uo15pZ*_!O
]YD;#_o9U5_nk.,rr;Kf!q1:pm5d$c5t4LE<E`a?Bl8!9I"@$0Ng#j#TVSNfYdCpP^V\"4bg4Ya
eCN7(f@A<qahk^!X.c$^Qm.reS"-">9)o!$USXidW2Zi!Xf\b0Z*L^B[C*HP\@K2^6iI&l`8'8+
g=?N0!.t3grkn`Prk\WMs1eQIs1JLb!!#Rc]`#PD^Akq[^qmk(_o0O5`lH0BaiMNBkl/g+!V(O/
3[m-U7SHZ[>[_5[E--ASJqo8FPa@l3USb&nZ*UpN]YD;#_o9X7_nt:/rr;Ng!rgD8m5cRU5!qn:
;cd76B5;I/H@LR'N00EpSt`-`YHkXK]u%_/bK\D^eCOWN51S'+c.(S1k2"=lR>6?HLPh(POu!(<
@%3^ZTq\?ZVPpJoX0&P.Yd(L?['d?OX9JY)2U'h2iSiZ)!!$QKrl"fRrkeZNs1nZLrk8QLMZ<_Z
ZM_B6^&GbG^]2(\_8=(,`5T^8a2l?DaMu9grlkBCnU'mh.6Ba_8l/Ml?Y*tjF*N+aL5M"TQ^aSA
VQ-c&ZaRHX^VRh+`Q#m8_Vb=ks7-*joohK)3\NT\7S?QX>$kfSD/siJJVK&BPaIu6V5^T%[CNrb
_oBgCcdC4lf%AU)d*UeAqu6Tp2?*L<_ME)JG'ngn1'R_nPEhN(S=ZCIU84W`W2co"Xfnt7ZEpr-
!!!5.`mWZ%jlIZd"D,XQ`;RUQ_Z%CO_#D+J^&o2P!*f94rkAEIs1nZNs2+fSs2>;aa2c9Ba2Z-f
rlkBCn9X[e-pC$j:K1M)A8,q&G^Y*sMN3jcR\$:OWi`M3[^s)c_8F41`l5m6jSo/Ync&X=?L0^(
5!he7;cd44Anu@.H@CL&N00EpSti3aYHt^M^;7e0bKeMae^rI*e^;n3r:p:9r8t/)?tO>"H[gX"
MN!UZQ'[r/SY)UNUnt#hWiN8*Yd&TT!"Xgr_TUNhkNgIUhpork`5MSkrkeZNrkSNJ#ekZd!"$*J
]tO<Ws1eQKs2"`Ps2=rV$-($+aN2B?`p`oNrrMG*o/\$X6V'sN=^GTOD/siJJ;&i>P*VQ.U8Fol
Yd:dL]>)2"`;[dV_u]$"rr;Qh!rSrcm5cd[5Xe:@<*EX>BPhd6I"@!/O->s$TV\WiZ*h-U^r+48
c-Xngf%AU)e`-3!pAZ97p!8%d>\.eoG^P$pLl77TPa7]*S=ZCJUSXleWiN4drW!Kn[^j,ldGF'R
md0,ne'#_Kr58KMrk\TLrkATK&-)]Iq7cjCs1nWM&&Q6&`5T^8`lH0@`l,jarlkBCk\9WL+Zr(^
9i>)!@qTXuG'e^mM2dX_R[p1LWN<>1[^s,d_8F4j`WO/ri;NZTo)AdYE'3MK.Qg$f9N"qs@:j:o
FaALhLl@I]R\$:PX08h<]"Z&#aii#Xe(*(%e^s-cnc(?-O%rJP?Y=;"H@CI"M2[LYPa@f,SY)UN
V5C2j%KHJ3UpRVC_91Elmd]hpYhmjN`V[RO_Z%CN_#D(OCB+>G^&5VC^])%K_>h@P_u@RU`Q#ps
`rsB"_s@<GrrM1ko/\9^6:XaJ='T3ICiOWFIY<N9Od2?*U8=ijYd:dK]YD;#`5]j;`5(.+rr;Ti
!rgP@m5c^Y5=J.><*<O;BPhd6I"@!/Ng#g"Tr"`jZ*h-U^r4::c-b"if%8L(kPOcK*<,&X:(I=)
:Mb#oH[^R$M2dRZPa7`,SY)UMV*t:_&mhfK[CEogdc'TN!!!Mlh9a6]rPSNLrPAKK"i%YS!!%cL
^])%J_>h@P_u@R]`Q#p<a2Z*9_<(X>rrLkXo/\9]5t=UG<a9'FCN4KCIY<N9OHl6)U8=ijYd:dK
]YD;#`5]j;`PUO2rr;Ti!r/HPm5c[Y5t=UG<a/sCC2\6?IY3E7OHl9+USk/rZa[Q\_T'^AcdL:m
f@J[Cr9aM%lC&@d!'4M"CiOWFItEE3N/s*cQC455T2PXg'd]o_Z*UmM^WP9pnIGU"fB:i4`P]Uh
_Yq=K^]RO9!%Ri[rkSNLrk\ZQrl"iUs2G8``l5j4kPi^*!RFX73[d'T7SHWZ>@;#XE-6JTKSYPJ
Q'n2;V5^Q#ZaREW^VRh+`lH-=`p<X's7?6ljD6jE3[m-S6V1$O=C,KMCiOWFJ;&i>P*_Z2UoCK$
[CNud_oKpFd*pOrf%KNim/Iq%m5t@T0iM:bBPhg7H[g[&M2[LYPa@dkrW!]nU8b2sZF.3S_pIE2
2&AF%mc<!B`P_Jfr5&BJ!3#ks!20;*r4r<Jrk\ZQrl"iUrPeuZ_nb=2c2RkE_+F5Z+ZhqY92JYn
?tF+mFa8CfLl7@[R@KtIWN<>1[^s,d_8F4j`ra6%n,<:XrrKl-m5cXW5t4LD<E`dABlA'<I=d64
OHc0(U8FrnZa[N[_Ss[AcdC7mf%0'_lMhY!a`]g37o38oASZ7,H$t6sLPh(QN>DKn(FboNVQ$]%
ZaREYaP,nFp%J(Mleg(0_tV%F_>h:Q1]RMJq7usFrk\ZQrP\`Ts2G5]`59:Fr6>6Crj5)b3[[!R
78$HX>$toWDfg8QK85AHQ'n/9V5^Q#ZaRHX^VRh,a2l<?bOb]4o`#!fLck&c-Tsge9i>(u@V9Ir
FaJUkM2d[aS"HLTXKf.C]YMG*b0A>_e^i7.q<7hmr6f+<4[i(F>[qGaEcubXJq]#,)?'UQ#a:jR
Uo18rZF%'O^sCs-o_8+Rn`\ZM`:q.G_>h:RNrT.a\,3u>_>V7N_u@UP`<=)q^Y/M/rrW.o5k@#a
4$Q/+:K:S+ASZ4+H$t=#Mia3lS=udYXK]%>]"Gep`5Td<a2Q9qrVuNi!U+"[3[[!R77g9S=^GWQ
DKC#MJVK&BPaJ#8VQ-f)[^s/h`Q??MdaZgtg%OS.')JdZ'g4oX;d3[BCN+B?I!e0^rW!iHO-lE.
Tr"`iYHkRF]>iFdm(fduo^:\ea2@YgqS<<L'EA,Ur4r6HrPJTPrl"`R"N/'hd.klc!rJ-DniA*^
77p?U>$toWDfg8QK85AHQ'n/9V5UK"ZaRHX^r"%/a2l<?b4Y`5p&>*lb=q:a-Tj[a92S_o@:j:o
Fa8IhLQ.F]R\-@RXKT"@]YDA)b0JA_e^`7=roX4koa:a"*BZY[;-7.8Bl72nr;[i=3KB5dR%'_B
VlHo)ZaI<Wah7ZS*r,I#l.aJ%_Z%CK_YM"OBE/#R^qfc^rPJQOrl"fT"iS9o]uBTfrrMh@nN&!]
6qU6T>$toVDfg8QK85AHQ'n/9V5^Q#[(!W[^r"%0aN;NBb4Y`5p&>*eU-S*,-9adf9iG,!@qTUt
G'eanM2mddS"QUWXg,=F^;7b/bKeMbe^E1IjSoX#!!"?,4=2[2,8hL2qu@cGBnqV,Od2<)TV\Wg
YHkLB\\QGm!!)H]p$^qha2@hkr5/<H"L8(/!292(rPJQOrPSr[`5KO/]tj'\rrMP%nN&!]6qU6T
>$toVDKL/PK85AHQ'n/9V5^Q#[(!W[^r"%0aN;NBcM@G=p&>*RD`dAJ-9FI]8l&Gj?Y*qhF*W1c
LQ%=ZR@U(LX08k=]>)5&aj&2\eCE:Ej8T>]7S-ZB3Wo*V+"V_gK8,8CP*MK-USb#mYHbC@]#Q'G
;;h3Bo'4rP_th+K_YCqL4TGIQq8*!GrPSNN"N%sf]][Q>!T-lH3[R'[8l&Jk?tF(kFEr:eLPq7Z
R@U%JWNEG4\%98h_o9X:aiMHNr;HZgrrL>Cm5cRV5t4OF='T0GCN+EBIY<Q;Od;K/Uo:B![CElb
`5g$GdF6UtnG_GI"TIbZ75juB,6T&=D0'oKItNN7Ng#j"TqnTeXfo"8[_C#GnFcPMo^:Ya`UCVG
OoPIc\,*r=_YqCM`!+&q^qIA=cMmtFa@Gb]-9jpk:KCY+ASQ.*H@:F$Mia3mSYDs\Xg,7C]"Pqt
`Q$$AaMus4rVuTk!r],dm5cOW6V'sN=^>NNCiX]HJVAuBPF%f5V5^W'[^s2i`Q??MdaZh,r87N,
j**Bt'FkBX*^X%=ASQ+'GC+jmLPq7ZR@g4MWN3/)Yd1dUe`cStp\"(=f#bf._",;D('">UqSE'G
rPSWQs2,#V^qI81rm1TGqQ)pP3[[!R77p?V>$toWDfpASK85AHQ("8<VQ-f([CEi__8F74ai_`F
iVrfUpAY3fIlm'Z,s+@\8l&Di?Y*qiFEi4cLQ%@\R@^.NX08k=]>);)b0A;^eCsHLs"=5p<BE6&
4$cD5<E`a?Bl.p8H[g^)NKTWuT;/<`X/rJ.ZauN<<qH%Xo'5&U_u.=B_#nHQ!$1pPqni-G"i7se
\BXrtrrVbO47YE[4$Q/+:f^b-ASZ4+H@CL&N00EpSti0_Y-PIF]=u,!`lH3DaiEQCrVuTk!T[SS
3[HmT7S?TZ>[_5[E-6JVKSYVNQ^aYCW3!83\\,_raNMlVe(!FSh>[QMO[K+/.Qp*j:fU\,ASQ.)
G^P'sMNF'kSY;mXWN3,'Z%rA[!V#aRoBkJ_`507(rkS3C"LeI5!1s)(qni0H#/\3k]strSci4%0
G4@U?4$Q/,:f^e.ASZ4,H@CO'N00EpSti0`YHkUI]YD>%`lQ<FaiNfGs7cNprP1ni3[HsW8PW2d
?=RYcEd)k]Ko1nTR@U(LWiiY:]"c,%air)ZeCX?lh#@E4ErA$<4@;_;<EijBC2S-<I=m?8Od;H.
U84ZbX/rJ-rW!51nFQ8>j4),crkJEIp;$hl!!#jm_>;%G_uIRW_8![n]'muF!S(NH3[QpR78$HX
>@D,ZE-6JUKSb\OQCFMAVlR#,[^s/e_SjI8b0%iGmJR"YrrW%i5OLQY4$Z5,:fU\+A86%(H$t="
N0'<nSti3aYI(gO^Ve+7cHt"hir.X3!rKPtrAl8h6q^?W>[_5[E-6JWKo1qUR%0eCUo(,lXKE$@
.^81,o'YVkai"%krkS9E"H3<\$GHARq83*Jrk\lT^V.1diVq7)!rfl/nN&!Y5XnFE<a9'FCiOWF
It``=P*_Z1Uo18tZa[NZ^r+.3ai_fKbk1i5p\t<dJNW?],WnC`9Mnhq@:j:oFaALhLlIO_S"HLT
Xg,:E]YVP-bKeMafC7Jn!q:M!r]2Ji5=J7D=']9JD/slLJqo;JQC=A;U84ZbX/rM3][#BumHirf
b/;!.^AbnH_"b_K^'t&BL&8^^_Z.OQ_?[om^V.4dcM>lf!rSBNnN&!Z5t=UH='T3ID/j`GIt`c?
PF%c2Uo:B!Za[NZ^r+.4aihlKd.mP=p\t9SAaMN(5!qk9;cm=8BP_^4H[pg-Ng#j#Tr+imZF7?Y
_SsX@cdC1sq:YcLrPDM33[QpR7nloa?=R\fFEr@hM2mddR[p(FUo(,lXg#4LdGX3Oj4VYp^\tnH
^\ttD_#md>!''kloYLjL^q[Ro[D;=[rrVh347P?Y4@2P4;HR46BPhd6I"6s/Ng,p$TV\TgZ*_!P
^;7b-aND]La4:(bs7lTqrlS%)3[HjP77g9S=^GWQDKC#LJVT/FPaS)9VlR#-\@]Mn`lcQQda7(R
f`)$CN^*J',s4Ib:/kD(AnuC1I"I-4OHc*#SY)XPVl?_^CI>ndhW!JFbJ_-o]`>eG^\ttE_$$G3
!!B\-_=YSM_8!^s]!f$*rmLfIhGE]&-9OUc:/b;%@q]b#GC>$sMiX*jSY;mZXg,7C]>)2#`lZBH
bJi-3r;ZQl!rK#bmQ)UU5t4LD<EijBC2e9?IY3H8Od2B-USt8u[CNrd`5g'HdF$;/fDbpObuag(
,X"Of:f^h1BPqm:ItWZ<P*MB'SY2^QVl:F3!0tRPg=k#u_nQ)_!5AHIrP89G!uV(=H2GGO_Z.FW
^V7@l[]n!Oe,KLJ\NfRE-9X^f:/b;%A8,q&G^Y-uMia3lStW$\Y-PIG]YD>&a2uKIb/<0:r;ZQl
!qhX<mQ)UV6:X^I='T0HCiOWEJ;&l@PF%c3UoCK%[CX)h`Q69LdF-bEf)Gg@LckVs,We4Z9N"u"
A86(,H[pj.O->irS"64IV5H?S!,9-dcI'qX_S,r]!kl=^rkJEIqnWAH!!"SH_>(nI_$Iik^:_%d
Z+02MrrMbPn2_pY5=A+><*EX?C2\3>IY3E7Od2B,U8P#pZaREW^r"(3b0.uN`9R@"s7lTpl?;SV
,s+:Y85<)c?=RYcF*E"_L5V+WR%9tKWii\;]>)5&air)YeaEFfrrW/@9)L^p4$cA3<*NdDD0'rM
K85DHP*MH)StVpVB/X%\[_0Q"aMbm-rO`3F^:sT]rP8?I"KDG%!OMX=qSN6J$,F<i]XkSXY1*@!
rrLnbn2_mX5=J1@<a/sDCN+EBItNQ:P*VT0USk/rZa[NZ_8F:6b0/#Oc1Uu6q#:BL@dZ9&4$Q/+
:fU\,ASQ+)H$t:"N0'<nSti3aYI(gO^r4:9c-Oehp!j!Bm:cAX,!A:d;-.(7C2e<AJ:rc=OHYuu
S=cLMVPpPtZ+%E`_nj+#rk&3C!kl=^rkJEIr4rI[!!$s4_>V7L_$Rol^:_(fZ`Uq9ec,^L[S$$M
-9OR`9Mnks@:sCrG'eamM2macS"QRVXK](@]"Z#!a2uNJbf\Q7qu?Kl!rg#2mQ)UU5t4OF<a0!E
CiFNCItWW;P*_Z1Uo:B"[CNrc`5p-Id*^nPeGfUKTL/Q;+ZhkW9iP;(Ao)L3I=[-1NKKHkR[g"E
Uo(,lY-POK]tV/W]*>p]]tV4q^\knG_#nWV!$V*Q!5eWL$GaEj]Xt\\Y-$jcf)GgCKK\fc-9X[c
9iG,!@VBV!GC4sqMiX'hS=udYXg,7C]>)5$a2uNKbKJlDqu?Kl!r],emQ)RV6:ajL=C#BKD/j`G
J;&lAPF.l6V5^W'[CX&f`Q66Jd*qI3rrLDLq`5c\6V(!R>[hDbFF&CgLPh+SPa@i0TV@BJ0m9b+
[^l=IrjrBI]tM.o^VB`_rP8WA!<<,!o"k@A$,=3e]=>8QWQP"frrW2N>OOU"5"%t;<*<O<Bl8$;
I=d33OHc0(U8FrnZ*q3U^V[t2b0/#PbN/["s7uZrqiF#13[?dO6qL0R=^GTODK9rKJVK)EPaS):
VlHr,\%BAk`lZHMd+7p=rrVVF48V&g5=J7E=^P`UEHce\KSYSJP*ME(StT!]!(EumZF.0O\@K/]
]=Y_g]tM/Z^\ttH_#l7h!+tf<s1nZK#e[d[[BZd;jSmg3!r8Z[n2_mX5=J4A<E`dACN"BBIY<K9
Od2B-USk/rZaRHY^r+16b08,Ram&[%q>UNkG<>7S,We7[92JVl?Y*qiFEi4dLQ%=[R\$:PX0Aq>
]>)5&air)Xg\T@u!rKVuq`6;k5tFdO>@M8_FEi4cKo(bMPEqW,Sp158)QU!EZF%'L\@B)\]=Y_h
]tV4q^\knH_#nu`!$:aJrkSQJ#e[d[[^*$5`VJ0d!p+8\n2_mY5t4OG='T3HCiOWFJ;&f>PEq]2
Uo:B!Za[N[_8O@8bKS5ScLUi3q>UKc@I?0%4@)G1;-$q0Anu@.H@LU(NKTTsT;8EeYdCsR^r+48
c-O_rrR1]Iro.&V3\!9\92\kuASQ1,H@LO$MN3g`Q^XJ71hE)SXKJe6['d?O\@K2_]Y(qk^AbnG
_#D.QZiU4-Wq=.-^^Rul^:_(h\$N99WTF-5rrW278+/Jc5!qk9;d!C9BPqj7I"@$0O-H$&Tr"`j
Z*h*R^V[q0b0/&Sb1c^gs7uZqjA52),X"Ia9i>)!@V9LsG'eanMN3meS=l^YXg5@F]thS-bK\A\
mJbN5!qpk#q`65h5Y"RL>@D/^F*E"]KSYPHP*ME(StVpVW2cr%Z*LaE[^WcX]">Vf]tV5[^\ttI
_#l4g!,M5Cs1f&X^:q7m]!o&NWhR3!gA_6HLcb)e-TaR^92JVm?Y*tjFEr:eLl7@[R\$7NWiiY9
\\,_q`Q63Gc-=DnrqcZjrrLV;mQ)RU6qL-Q=^PZPD0'oKJVK&CPaJ#8V5g]([^s2h`5p-Hcd_U6
rrW1m5l*Mn4[_q?='fEQE--ARJVAr>OHYuuS=lUPR&R-gYHbF@[C3TU]">Se]Y2%n^VB`_rkS\Z
!!"89_>h:[^V7Fp]XkV\Z)jdug]$".!rgJImlDdX6:XaJ='T0HCN4KCItWW;P*VT0USk/sZa[Q[
_8O@8bKS8VaRAs)qYpWq]KGXD,We4Z8l/Mk?tF%iFEi4cLQ%=[R@^.NX08k=]>)5%air&UfD3bm
!m=D#q`65h5XnII>$tuZEcubXK85>EOd29%PpZe:UT1AtYd1UB[^WcW]">Vf]tV5[^])%J_$%"C
!!0Cu_>h:\^V7Fp]XkY^Z`^:#Y3u;FrrVbK5O^]]4$H&):K1M(@qfh%GC4srMN3pgS=uaXXg,7C
]>)5%aND]NcHF<%q>^<k!W/ka3[?gQ7nZZZ>@;#XDfp>SK8>MLQC=G@Vl[)/\%BDl`Q??Lc.r6C
rrV.g3Vkcg4[Vh==']BPDfg5OJV8i<OHYuuS,rZj@&^3/YHbF@[C3QT]">Se]tM.o^VBc`rP8RZ
!!%B=_>h:\^V7Fq]Xt__Z`gF)T??rcrr`8L>!)47-TjX`92S_o?tF(kFEr:eLQ%=[R@U(LWi`S7
\\5bq`lQ?IcHaV`r;$BhrrVt_3U]!S4[MY4;HI+4Ao)F0H@LU)NKTWtTVSNfYdM!R^r+17bg+Mq
rm1TGms7"j3[?dP7SQic@:s@pFa8FfL5LtQPa@gU!!#9^WN3/*Z*UgG\%'!J]*5j\]tV7rrkSKK
"(VB33;(,f_#M.Y^:h1k]!o&OXf%heg&Bn/!qD:6mlDg[6V1!N=C,KMD/j`GJ;&i?P*_Z1Uo:Au
ZaRHY_8OC:bK\>XbiA]us8)`so8#Eq3[6gU8PN,c?"7PbEHcb[Ko1nSR%0kHWNEG6\\,_qa3)WO
d-f3Z!r%j>q)U&i6V:3W?=R\eEd)k\KSYPIP*MGC7[9dHW2cr%Yd1XD[^WfY]=Y_h]tV7rrkSKK
"J5Yo"hjWGrkK#Y^:h1k]"#/RYH"@dXmb3J!rg/<mlDgY5Xe=B<EijBC2\0<I=d33OHc-&Tr"`j
YdM!Q^V[t2b08,VcHbkQs8)`smWpn_3[?dO6qU3R=^GWQDKC&MJVK&CPaJ#8V5g]'[^j)f`5g$E
c.)F3rrW(t55@5j4$Z;2<*NaCCiFNCIY*<2NKKHlR[p(FV5L>pY->1;[C3QS]"5Md]Y2%n^])%H
_#k,H!0I2sq8*'G&&5il]t:kd[^33=Vk0@"qVhGUms.%a3[[*Y852ua>[h>]EHQSVK8>JKQ(";=
VQ-i*[^j&d_oBgAbg+PZde3D8qYpTWD!sD14$Z5-:K:S+ASQ.*G^Y1!Mia3lSYE!]YHkXK^;7b.
bKS2bqTf*Brkh2"3\EQ_8l/Sp@qfh$GC+jnLPq4VQ'e)5R"M98X/rJ/ZF%$J\%0)^]Y(ql^;%Fu
_#2"N9)nrhr5/?IrkK)[^:q:m]=GDYZEC1%S"\[:j8T2NU-\6/-p0aa92JVm?Y!nhF*N+aL5M"T
Q^j_EW3!50\@]Jk`5g$Ec-O_]jo"oRqYpTN@IH6&4$c;/:f^e.Anl:,H%(C$N00BoSti3aYHkXK
^;@h/bKJ,er6G<Drmam@3[d'U7nluf@:sCqFa8FfL5LtQPa?b;!%t+HWiWA-ZEpsI\%0)^]Y(rW
^&l'b_#2"NMZ<_jqni6Hs1nZL%_fWh]=PJZZ`gF,T:!"RjSo;\aBS*o.6B^\85<)c?".D^E-6JV
K85DIQ'n5;VQ-c'[CNrb_o9^>bg+S]dI@/1s8)`rcrf`t+$DnZ9i>%t@:j:oFaJUkLl@I^S"HLT
XK]%@]"Z&#aSs?`ir..%!TIG[3\<N`92\kuA8,q&GC4poLl7:WQ%o>>"J2KiX0&S1ZF%$K\@K2_
]Y(ql^;%J!r4rNK"98FuqSN3Is1nZL&&,`i]XkV]['?a4UR[abq;qVXl?,;Z3[d-X7SHZ[>$tlT
DKC&MJVAuAPF%c3Uo:>tZaRHY^r477bKeG\cIM[`s8)`rbZ4*m,X"Ia:/b8#@V9LtG'eamM2d[b
S"QUVXKf.B]>)5%aNMcLkPia+!pXM^pc9uh6:amQ>[h>^EHQPVJqf/BOd)4T"#>q7W2m#&Yd1UC
[^`lZ]=bei^:q@t_#;+L_?2On!,_PKrPANL&\l&n]Y(ke\$`NGX/;SYRFqE"rrVki7.3/c4?u>,
:K(D%@qTUtFaALhLQ.F]R@U(KWNEG4\@]Jl`Q63HcHsq_gAL^FqYpT4:[^=h5!qn;;d!F9BPhd6
I"@!/Ng#g"TV\WiZ*_'R^V[t2bKJ#mbl7b7Gs(j_/NZ3d9iG2&ASQ.*G^Y-sM2[LZQ@#$lU8=cf
XKA_4ZaI6N\[oDc]Y2%n^VIY`_>h@SErZ1npqm$HrkK,\^:h1k]"#2TZ)t"$SsQ5-rosFaq70Gm
3\!3U6qC$M<a9$DC2S*:I"@!/Ng#cuT;8BcYHkUI]YMG)aihrTd*Ku$rqHHhrrK`*n2_gV5=A+>
<*<R<Bl8!9I=[-2O-H$&Tr"ckZF.6U^r+.4bKJ#rbl7b9J3EWf/3Q?m:fgq4BPhd5H[g['MN3ga
R%'\@Uo(,lXfo"8['mHR\\#Jd]tV4q^qfrcrk\f9!!!2:pqm$Hs1nZL&A>`h]=GDYZEL=,U7@LL
li-5M"8g5P4RY<[5!hb5:fU\+@q]_!G'\XjLQ%@[R@U%KWNEG4\@]Jl`5p*FcHstadJ!;6qu6`s
Yt4@F+?MhW8l/Mk?t="jFEi4cLQ%=[R@U%KWiiY8\\,_q`Q8&%!;4S9!r%^>pGsie5tFaM>@D,Z
E-6JTJVK#@OHc*#Mc">fW2cr%Yd1XD\%&u\]=bhk^;%J!rk\QM"V:Y6PJb<6_Z%@b^q[Xu]tCtg
\@/cLY,\=kR?FlEm/I1ddq(]23\!9Y7S?QW=^GQLCN+B?I=d32Ng#g!T;/?cYHt[K]YMG(air#U
d*^.srV$9grrW.n6h!,^4@)G1;H6q/ASZ7-H%(C%N00BnSt`-_Y-YRJ]YMG(aNM`Pprra>q3F;?
3\EKX7SH``?=[bgF*N(_KSYPIPEok^!%=\AWiWA-ZEpsI\@K5`]Y2%n^VIY$_>M.P7fWNrr58EK
rkT;a^V7Cp]XkY_[Bd!;N^tK`OgOKHn,ELeb]#]53\!6X7S6HT=C#?HC2\0;I"6m,NKKNpSYDs\
Xg,7B]=u/$a3)WOd*^4fme-VWrrW+_5k$f[4[MY5;H@%2Ao)F/H@LR'NKKNqSt`-`YHt[K]YMG(
aNDZQqTSs@qk6CN3\<K_8l/Po@q]_!G'\XjL5V(TQ&,MA!L'F\X0&S1Za@0M\\#Jd]tV4q^qde&
rke]Q"+L:N-MkXX_Z.IP^_FPt]tCtg\@/?AY@@P4/!Gsc%uC5Urr`2LC-V/J.m-*d8PW2c>@:uU
D/sfGItNN7O-H$%TVSKeYHt[J]thP)aihrTdF$=pp\"R`rrW(]5k$f[4[Vb7;cd76B5DR2H@LU)
NKTWtT;/<bYHt[K]YMG(aNDZYr650BqkljU3[d*V7o*,h?tO1nFa8CeKo(eOPa<pUC7_<%X0&S1
Za@0M\@]Ac]tV5[^];4L_Z.LVXoJG)^\YkG_Z.Ff^q[Xu]Xtb-+%*B=P5kS8Rup/?!)!7`rri>o
]ja(l3\3E[77g6P='K$BBPha3H%(=!MN3jcR\$7NWiiY8\@]Jl`5p*Fcd:+fdH^]'s82ftqiX56
3[?mV8PW5e?=RVbEHZ\ZKo(hRQ^aYCW2m/0[^s/f_o9^>ajp@6rrW(q7.rYm4$Z80;d!I<C2S*:
I"?s,N0'6hR@KqFV5UDqYHbF@[^WcY]=bhk^;.P#_>_=M_Z^SO!1:^CqSW9K(r=#'^V@IpM#[MY
TX(ahC5ALL/H>c>l2LGW"T@GZ=?Q(6/NZ*]77p<Q<a/p@B5;I.G^P$qM2dX_R@KtIWN<>1\%98g
_oBgAcHjncd+7m^s82ftqiX563[?pX8PW5f?=RYcEd)n^Ko(hRQ^aYCW3!51\%B>i`5Tg>b2*'@
rrW(q7.iSj5"&%@=']9KD/sfHItNN7O-5fsS=cO<VlHf#Yd1XE\%'&^]Y2%n^qdec_Yq@T`'FA'
B`,gH_u7IP_%sh`^:h03!!S%_Z*(.+US!l>:OSS^p\Ommrq4BT7.3/h4?l/%8l&>d>@1lQCN"<=
H[pd*N0'9kS"HIQX0/b9\@fSn`5p*GcdC1gd,Y)ss82ftq1h0(3[HjP77p?U=^P]RDKC&MJVAuB
PF.l5UoCH"[(!Z\_8O=6ai`TAbl7bAVaU2=/3Q<k:fUb/Ao)I0H@CI#MN3jbR"-[g)Q0R8Xg#+;
[C3TW]=bhk^;.P#_SZ8hs25"a!!"_K`;[[Q_BQ]p!IDJnP$bN'[Bm-?Wi)S^Q&C-5H(N4>n+H>?
j1pRR54CTd4$>hs85)iZ=Bf-BB52@+GC+glLPq4WQC=D>V5^T$[(!W\^r478bg"M^da?S6oDe^f
!r%1-nN%sX5Xe=A<a&j@Bl8!:I=d33O-H$%TqnZhYdCmO]tqV*aN;T`rlkBDqkZ^R3YX\B7o!#e
?Y+"lFEr:cL5CnOPaGVQ!#1WrX0&V3ZaI6O]">Vg^;%J!_8=(g_u@RWVZ6]%_th7L_uIRr>6"X:
ZbF)a]"#2UZ`gL2VP0NEML]YU@;^ORN-Jl(4RbBd4$5bp7S6BP<ENL6@qTRqF*Dt\K8,8DP*VN,
Tr"]hYd1^J]YMD(aNMiSdF-Iih>-[Bqu6`nM*LMk,s+@]92JYn?tF(kFEr:dLPq7YR@KtHWN<>2
\%B>i_o9[;al*0BrrW"`61m8h4$Q2.;H[=;C2\0<I"6p,N0'9jRKNTjBrJ)7YHkLB[^`o\]Y2%o
^qmk(rkn`Rs2>'J!!&bk`Vd[R_]Qb'('#PY^:h1k\[])TZEL@/V4j?@MLT_F3DW.34Zi-o1HR`a
6:FI>:Jt4s?=@DYCi=B=H@CI#M2dX^R@KqGVlHo*[(3f__8OC:bg+S_e'c\1rU^'drrW([5k-l]
4@)G0:fgk/Anl:-H%(C$Mia3lSYDs[Xg#1A]=l%u`5fs?i;Ut#!r.mCoK"Ne6q^BY?"7PcEd)k[
KSYPIPEqV+5+AFJX0&S2ZaI6O]">Vg^;%J"_SX4.rl"fT"#g2[G59,W`$N)`XM_o<>ee(h^:h1k
\[SuQZEL=-V4a6>I3S_!@96/l5s7:kpH&=91d!od6UjU?:Jt1p>@1iNBl%a1GC+diKo(eOPa@l3
USb&nYdCmN]thP*aihrTdF-LlgA(7=qu6`pP!JOu,s4La9Mnkq@:s@pFaALhLl@F\R@U(KWi`P6
\%B>i_o0U:aP[!@rrVqU5P-ug4@;\9<EipFCiOWEIY3E6NfoZqS=cOPVlHl&ZEpsI\@]Ac]tV7t
_8=+-`5MVn"NG6X!(?e*r5E<S*<7V)_o'@._8*h"]tCtf\@&ZJY,nV!T:1cJ!!#i0@9HN&7mfO+
5!;%k4Zttn5XIh-84lTQ;cQq(?=7;VC2S$6G^FplL5LtQPa@l2U8FokYd:gL]thM)aNDcRdF-Ll
dc^W#s82ftqiX573ZpOO852ua>[h>^E-6JVKSYSLQ'n5<VQ-c'[CEi^_8F4j`rY&rbl7b:K0Jog
00;Hi:/tM,Ao)F/H@:F#MiNscR%0bBJmbU[YHbFA[^`r]]Y2%o_84"+_o0L4`W!j[PlLe$a8<mV
`<;;*!!&&Y_uIRr_8*h!]Y(hd\$WKGY,eLrT9qqF!&\k8AmeeB:J=AO7f>b77Ros?92//Y;cQq(
?!q/RBkqU,Fa8@aK8#/AOHc'"StVsYX08h:\\#Sl`5g!CcHt"ge'Zk@nc/Ld!rJ-GnN%sZ6V'pL
=C#BKCiOWFItWZ<Od2B,USb&nZ*h*R]tqV)`lH-UrlkBDn97"e3\EKY7SH``?tF+lFEr:dL5LtQ
Q'e&/%fc^$XKJh7[C3TW]=bkm^qmn)_o0L4`W!j\a1VKR!PeWSr5JiXE!VO(`;daT_BHb2^V7Cn
]",;XZa$^8W2-#%!!"-lH?=(=A7&PC<Dudmr_PLM;GpIo='8a5?XdPYBl%^-FE`%ZJ:iW7NKKHl
S"?@MVlQu+[CEi__8O@8bKeG]e'ujokParNqu6`pP!JOu,s=Ue:/b;$@qTUtG'e^mM2d[`R\$:O
X0/_8\@]Gj_o0R8`7"S-rrVFu47kQe4$H)+;HR::C2\3=I=d00NKKHlS":o+!)fo$Z*UjH\@]Ac
^;%J#_Sa:0`Poj:rl5Df('"?)aN2EB`)LFL`W3";`W*mV_uISV_8*h!]t:kc[^<?FY,eLqMZ<_c
Lk:&"Ec,W'@UEDF>$5!1=]np4>[CcG@qB:eCi467G'J@_JV8f9NKKHlS"?=LVlHo)[(!W[^r+15
bKS;ZdaZdpf_4h7qu6`pP!JOu,sF[f:/kA&@q]_"G'e^mM2d[aR\$:OX0/_8\@]Gj_o0R8`ROk1
rrLqWniAEf6qU<X?=R\eFEi1aKo(bNPa@l2C]lt_XKJh7[C3TW]Y(tn^r!t+`5T^8`lH.!a9+F&
!.4^d"iEM(!.W&g`r<sV`$3+9_SO%&]tCtg\[JoPZ*(.*R/d3mOGejDI!9XNCh[NnA,]s;@K0dk
@qB4bBkqR(EHHALH[UErKntYJOd2<&StVsXWi`M3[^j&c_SsR;bg+S`e'ujqk5FfLqu6`pP!JOu
,sF^g:K1M(A8#h#G'ngoM2mdcR\$:OX0/_8\@]Gi_SjI6`ROh1rrW2E9Cb,%4[Ve;<a9-JDKC#L
JV8l=OHc*$SY24EWiWA.ZaI9P]=bhk^qmn*`5KX7`lH-@aN;P7!!$!taTTF6!!#Xja8a0Y`W!gU
_]cn5^q[Us]=YV_[Bm-AWf@&Z&snc@M1gD,H$47KDf'9*r+[p$Ci+'/EH?8HH$b!jJq\u;NK99g
R@BeBV5L>qZ*UmM]YD>%a2uNMd*^=leC*+Fn,N:b!rJ0HniA*Z5=A+><*EU=Bl.p8I"6p.Ng#ct
Sti0`Y-GCE]=l"s_o9U5ana3]!r]\sniABd6:amP>@D2^Ed)n^KSYSKPa@i0TqnQcW2[&.[C<]Y
]Y2(q_8=+.`Q#p<a2n8%"NlQ$!%\&k"3J5(=8p;>`rO-Y`W*mV_]Zh4^q[Ur]XkY_[Bm-?HN4$o
S=#P&N/3.<IX??aF`VU$EA2ESFED\NH$XjeJ:`H0M2[IWPEqT*T;&0[Wi`M3[^`r`_8O@8b0A8Z
daZgreEQr#s82ftrLZFI3[QpQ77p?U>$b`RDKC#LJ;/rAPF%c1USb)pZ*h*R]thM&`Q#j?q9Ap@
qkZIH3\NT^85<,f@:sCsGC+jnLl@F[Q^XJ=UmFg^C9b+R\%0,`]t_D"_o'F3`lH0AaiXP)$*+%+%
D4p8b0%Esao9B\a8a0Y`;mjU_uIPM^qRLo]=GDZ['<p;!'m6KR[0.uN/34@JUW#oH?jd\G^4R\H
@($fIt*$%KnkPDNf]EhR$sP;U8=fgXg,4@\\#Sl_oBd@bg4Yae'upsh=p@;qu6`sW($8<-9FI]9
2JVm?Y!kgF*N(`L5M"TQ^aVBVQ6l)[CEf]^r"".`P]Ucc2Rh<O7>7l4$cA3<*E[ACiOZGJ:r`;O
Hc'"SY;i^!!!n?['mHS]Y(to_8=+.`Q#s=aN;QHb0.uL!rr<I!$SoPb5TN]aT'<[`rF$X`#ut7_
SO%%]tCtg\@/ap!!$]/U7RjAQBI;fMMHn=Jq/CLJ,b'!JLZ;*KnbA>N/j!_Pa@f-StMjUW2m&)Z
aREV]thP*ai_iQd*gCne^W(6rpKg_rrKN"niA*[5Xe=B<Eig@Bl8$;I=[-1Ng#g!T;/<bY-GCE]
"Geo_o0O3_=,YK!paecniAEc5t4RI>$trYEHZ\ZK8>GIPa@i0Tqn4f!"kj1[^`o]]Y;1t_SjC3`
lH0Bai_d)b6Jmb!!*'#PKWsJrQ5&[s2Y,Yrl$)"_o'@-^qRLp]=PPG!!!(lWMcMbSsu.5PEClaM
i!7FL5,Y\/;Aq8Ll.+LNK93bPa7]+S=cOOVQ$Z"Z*UmL]=u,!`Q63FcHstfeCE'sh"pF<qYpT3:
@^Fm4?uA/:fU_,ASQ+)G^Y-tMN<sfS"HLSX0/_8\%95e_8=.0_nYE[rrUt\3V#3_4$Z80;H[@=C
N+HDJ:r`:OHZ$"SY;jWB6LG`['mKU]Y2%p_Sa=2`lH0BaiaS*s3(X0!!#q$KDWjkb5TN^aT'<\`
rF$X`#ln6_8*h"]tCs[!!"p]XfA=rU7\!FR$NquOH5E]N/R[ms,6o9.?0@KOHPfmQC+/3T:r$WW
2cr&ZF%'N]=u,!`Q-*Dc-OedeCE+!e*?r"s8)`rbuO<q-TjX`9Mneo?Y*tjFEi4bLPq4WQ^aVCV
lQu*[(!WZ^VRe*`P]FFrm(NE`Co_^/3?*d9N"u!ASQ1-H@U[)N0'9jR[p.KVlHl&ZF%*O]"Phm_
8=+/`Q-'Ab0'_,rlbVi70!<p[*6bkbPfT`ao9B\a8a0Z`W*mV_Ag>,^:gi\!!%AQZ*(.,Vkp/]S
su19QBd\uP*5^).?]mYPa.T&R@B_=TqeH^WN3/*ZF.0Q]Y;2!`Q63Fc-OeceCE."e)L2is8)`rd
ou<'-TjX`9Mnhp?Y*tjFEi4bLPq4WQ^aSAVQ6l)[(!WZ^VRe*`5B@<qp59Drhr0R3\*9W78-Q\?
=[ehFaARkM2[O\Q^aS@V5LArX\]M4\\#Ph^VRh+`Q#s>aiaV+rlbDf:]:=p50a&Mqoer\s2k8]s
2Y/Zs2G#V*5fY0^qX"P!#i;S['Hm;X/MqmU7e*KS!f\3Q^@];!1!PL.$p<gR[]h=T:hsTVPpMrY
HbFA\@T>e_8F75aihrTdF-Opf%&7%pAXFYqYpTOA+DZ.4@)G1;-$q0ASQ.*G^Y-uMiX*hS"?CPX
0&Y6\%95e_8=./_naFici4(EM`gJi0gJ0$;-.%5BlA-?J:rc<OHc*$St`'ZX/uI(!'@d(]t_A"_
o0O6aN;TJbKS2Tc-C\7!!4@icMl)fbkoT_b5TN^aT':&`l5p8_ns9h"98FH]=GDYZEUL5X/VtmU
S4?PSXl:@qO@MT-(UX"Tq\<XVPpMrXg#+<[^`o]^VIb+a2uNLcd:+ie^i=$dH(;ms8)`skANBP3
[[$U7SHZ[>[V,YDfg8QJVT/EPa@l2USb&nZ*_!O]Y;1u_o0I.`:D7S!q0eanN&Hf5Xe@D=']<MD
fpAVKSbYMPa@l2U8=fhXu-1Z!P#bU^VI_)`Q#s?b0%oNbg"DXcG).6!!!W1cMl)fbkoT^b5]T_a
T'<\`rF$X_\k',!%,Cm]XbP\['Hp=XK/A"VPU&_U7n6Qr1F"^-)%'.UnsreWN*#$YHbFA\@K5b^
VRh,aND]Ocd:+he^i=%db=E[s8)`smX.%d3[[$U7ncc\>[V,YDfg8QJqo8FPa@l2USb&nYdCjM]
=u(s_o0F-_!BAI!rgkLn2`Ki5Xe@D='T6KDKU8UKSb\NQ'e&4USaujY,;CLTt@tO^qmq,`Q-'Bb
0/#Qc-FV\d*TsA!!!9'ci25hc2Pr`bPoZ`aoBF&a2Z-<ErZ1AS%ueQ]XkY_[Bm0CY-"e*WMl_lV
5=!`s.ps<W2ZetXfen5ZF%'M]"Gbm_SjI9b0/&Ud*gFof%8L%dHpkts8)`soo(p&3[[$U7ncc\>
[V,YDfg8QJqo8FPa@l2USb&nYdCjL]=u(s_o'@,^?!Z@!rB5gn2`Wm5XnFE='T6KDKU8UKSb\NQ
'e)6USb#lYHb7=\\#Mh^r"%/`lQ9EbKS2TcHjHu^<k6P!<<,Dd*L&;cMu/fbkfN^b5TN_aT'9tT
`kDr=Mr"k^V7Cn]"#2V['Hp>Y-"e+WrB!qW;`b@WiN5'Y-5(8Za@0M]">Yj_8F42aND]Ocd:+he
^i@(eC3IUl2UV[!r8EJo/\9^5t=UG<a9$DCN"<?I=d64O->p"Sti0_Xg#1@\\#Pj_8=+,^qSY(d
Jj:<NB?Vj3]fSr9Mnnt@q]b&H@U^+NKTQpS=cRSWiWD0[C<]B@DBXq`5Td=b0.uPc-FY^4TH7pd
Er4"!5?h,rm1VirltDcq9/c[s2c2#\JYAI,f7L;_SO%&]tCtf\@/fPZa-j?YHIl,r2fn"s/uC+s
0<WO[C3TU]"Gbm_SjI7ai_iQcdC1ie^i@(e^NIOkl:MZ!W94n3[d*V7ncc\>[V,YDfg5PJqo8FP
F%`0USb&nYd:dK]=ktp_Sa7+^!cf*rrM(omlE'_5t4RH=C#EMDKU8UKo1nRQC4;:Uo15pYd1[G;
??i^_8F73aN;TJbg"GZd&[-G#E7'X'EA-<dJhMmci25hc2Gl^bPoX'`@q8l$aRdl`Pf[2_8*h!]
Xtbc\@8oS['R'Dq6Bq)s0EfS[C3TU]">Yi^qmq-`lQ<Hc-FY_daZguf@SU'g@bI/s8)`srkUbh3
[d*V7ncc\>[V,YDfg5PJqo5EPEqZ/USb&mYd:dK]=ktp_Sa7+]ug#rrrL>CmlE0c5t4RH=C#EMD
KU8UKo1qTQC=D<Uo:>rZ*UmK5l^m(_Sa@5aND]Mc-FY^d*\-k=T6_P*<6)9dF%s?s3L\irlt>aq
TAo^6N-s$Mo`u[`l5m7_SX.)^:h1k]",>\\$i]Pq6^./s0`uX\@K2_]Y;1s_Sa@5ai_fOcd:(fe
CE.%f@JI&m/?,Iq>UKH7.WGf4@2P3;-$t1ASZ4+G^Y-tMN3mdR\$7MWN<;0[CEf\^;.S&_SEk$m
aD+:riANV3\```7nll`?=[bgFaAUmMiX'gR\$4LW2m),['mNC">?Of`5fsAbKJ/Ucd:(edaGhUr
mM&t!!&V=dF%s?s3L\irQY2_rQ>1<rW!aIb0%iIaN)<>`P]U1_8*h"]tD"i]"5HH\Gj#S]">Vg]
t_A!_o0O7aN;WLcHjnde'uqIf*'[TeE6\ks7uZqkYh"5.6Kpe9Mnhq?tF+lFEi4bL5M"TQC=A=V
5^Q"ZF.3R]Y;.s_SX+&^?5FDrrW%N4RY<`4[M\7;d!F;C2\6@JVB#CPaIr4U8FokYHkOE]"F<D_
SjF6ai_fNc-ObaI<s9>e,Ii'-NF.;daHOjd*Vd<rm(Jepri_JrVupjrlb>a&';o6a2Z*;`5BL/_
8*h"^&G\E]DB,=]DfG_]t_A!_SX71`lQ9EbK\>ZdF-Lnf%8R,f@A@.puVPOrrM_&o/\<_5t=UG<
a9$DC2\3=I=[-1Ng#ctSYDs[XKSt<\@T>e^VRe(^qRJ9r6tZHmUGWG4$,Vn8PW5g?Y*tkG'edpM
ia3lS"QORWiWG1[CEf\^VRh$BZS<=bg"G[dCT\b+i)UWe^W(t!!&P;df.Yod/VGjcMPlabQ)2Q!
!5:*bl,`baqDh;aN)?@`Pf[3_SX.)^q[Xu^AGVC^&YqI^`1,,_o0O7aN;TJbg+P]dF-Loe^rI+f
@A@&p&=(Qq>UNlJNWQc.6Kmc9Mnhq?tF(jFEi1aL5M"SQC4;<V5^Q"ZF.0P]=ktq_SX+']?Brur
rW2D8FJSn4$Q/+:fU_-ASQ1-H\$p1OHl6(TqnTeXg,4@\\#Pj_1rX'IEKdYcHjkb=9&=Cchu<%-
ia7;e'c[ldEp7dci)/ablQ!frVusYaSj?]bQ#`baU-&/a2Z-<`Pf[3rPJBHrke]Qs25eo`lH0Cb
0/#Scd1"ee'uq!f@\a-e^N1>roF+TrrW+c5kI)d4@)G1;,pk/ASQ+(G^P'sMN3jcR@U(KWN<;/[
C<]Y]tV;!_SO"!_!okR!rAl\mlEcs5=J4B<a/sDCN4NEJqo;IQC=A=V5UGtZF%*P]Y;1u<WE+'_
T^<Od*^:`BaL]fe^i=%eKOtnUXRZ>dF$=fcd2U9prri\55kN_4N@<Drlk>as2tA`s2b5\!li3ur
5AHLs2=oU)TKh=aNDZKbg"J\d*^:jeCN4&f\"g,e(!^\j8\rT!r]GqoK"Ha5t4OE<Eig@C2S*:I
"6s/NfoZqSY;jXX08h:\%0,a^;.S%_8*^oir7L.!q1V#mlE*a5XnFE='T3JCiX`IJqo>KQ^aSAV
5^Q"ZF.3R]YD;#]c7ZGbK\A[dF-LmeAg&9edE2m!3jr#da?Ihd*M^:prriX1&_.R7EkSPrlk;`s
2tA`rlG)ZpW!3Q'$/2:b0%oOc-FY]d*^:jeCE.%rn%D+f$i.3r8RbPrrW2@;=ln!4?uA/:fU_,A
8,q&GC4ppM2d[`R@KtHW2m),[C3WX]Y;1t_SO%$\'Y!%rrLeZmQ*!`6:agL=C,KND0'rMK8>POQ
^j\CVQ-c&ZaREV^;.V(`hL;mbg+S^d_a8Ye^i=MfDjD/#ljt?e'cXkd*U,<cMYrcbQ4dVrW!$ub
g"E/cMu/fbkoT`b59<WaT'B]b5]ZbbRDbAcd0tcdaQ^qe^i@OfEKmXeC3OUiW&]Q!nq$Bof=9Z5
Xe=B<Eig@Bl8!9I"6p-NKKKoS=l[VX0/b9[^`r^]taQ_"Mh^]_!fkS!rTApmQ)p_6qL-Q=^P]RD
KL/QK8GVPR%9nHVlQu*[(!TY^;7_+a2uNKb:f%+dq]%QX8CKLf$XHs#g^rHdEp4ccHcC5qof)&#
6"T&@fFXIcMl)dbl#ZUb5TTabRVnCcHjkadF-ImeCE.$fDaD+f@JI$j8%X5q#:B[GP!pH4$Z5,:
K1M(A8,n$GC+gnM2dX_R%0hEVlQu+[C3TV]Y2(q_83q$]"-\rfDbpAK0AZa00;Be92JYn?tF+mF
aAOkM3!mhSYE![XKSq;\@TAg_SjI8b0-Og!!L'c/H>d1fDX>3f@I;/!"[N*dEp4bcHXW5bl#Wbb%
ptV!sk+7bl5lccMu5gcM5ZTbl>rfcMu5jd/VK*daQ[pe^i='f\"m0f@JF"j8%U4q#:EhPs+e$.m
$!b8l&Di?=[beEcueZKSbYLPaIu6USk,pZF.-N]">Vi^qmk&^:Uhrq:P]KrmP$<3[HmS7SHZ\>[
h;]EHZ\ZL5V.YR\$7OWi`M4[^j#a_8F74b0/"qrW!*bcl`;gao0j&f@JN2!!"_#dEg.ac-=JVqT
AoG,5qQC)lNYYrltGfrm12]pW`Z^rm1Sj$.72Le'lgse^i@MfEKjVe'm@QhZ*BN!rTGpof=Wc5=
A+><*EU=BPhd5H[g^)N0'9kS"QORWi`S7[^`o\]Y;1t_8*gt[^tc'g&D-MT0hs,5!;8(:/b8$@q
]b#GC>'uMij<pT;8EcY-PIE]"Gep`5]m@bK\@6!&;dKf#rs'f\+p/f@JJV!!$HScd'eZr6,#\!I
k@R!!,0prlb;brltGfs3LSfma(jYrmCbos3q"t"P))Sf@TuR#1qGVda@"Kh>d9M!rgJAof=Wc5"%
t<;d!F:BPha4H[^X(N0'9kS"HFPWNEG5[^Wi[]Y;1s_8*gu\?jZ\g&D-ACch)H5s%>"8l/Mk?Y*
tjFEr=fLlIRaS=uaXXKSq;\\#Pj_o9[<bK\>Zd>ZQ(f%&C)f\+p/f@AF#)ZTl"cHXSVbKA!,ao9
?^[4D4V!X!2YrlY5`rlkDeps/lbp<`Z`rR1\os4%)!s475%r7D5*f$r-oh"BJ&p\t<XBfkuK.m?
?m:/b;$@q][uG'\UiLl@FZQ^aS@VQ$]&[C3TU]=bkm^qmk%]XbD^n(RdCrP1bf3^5_n7SHZ[>[_
8]EHQVXKSb_RR%9tKWi`P5[^j&c_SjI9b08,VdF$FmeXn&%K\"f-f@AC"YQ+Y)_p$<Hb/sY(rl5
)[FpS'I!Z@ZPrlP2_rQ>2brlt8arm1JgpX/rfrR:er!7h)"s4@5%#M7PWeBu\#q:krDrrVhJ4S:
`d4[Ve9;d!C9B5DR1H@CL%MiX'gR\$4LW3!83\%&u\]=ktp_83n!]!\m"rnIGRm=ajf5WqJ*:/b
8$@q]_"GC4srMia3mSti3aY-PIF]=l"s`Q-*Dc-Obae'up'!WW6XI;j+TTu58Y!!#.(aiVTD`lA
"s!lV8mr;Zjt^&H%Oa8a3^ai_d*bQ#fbbm;P<c-FV\cd:&:dJMAie,@erec+/"fDO8,f@JO&e'Q
P)pY,ZArrW,$61m8i4[M\6;HR46B5;I/H$t:!MN<seR[p.KW3!83\@K/]]=kto_83n"]=5#\q:t
uOrkDA!3^5\j77pBV>@:uVDfg8RK85DJQCFPCW3*;1[^j&c_8F:6b08/XdF6Roet2sC"u$bq";N
h4PlUjbON5A[`5MVl!OYW"!!F%%_o0Lm`<O?&a2l?Db5TT\bl>ojcHjh`d/MGjdeqSleGn(uf)F
;!fEg*Ze^W$ne*-\^s7cNprmaj<3\<H\7nli^>[h>]E-6GSJqf/CP*VN,TVSKdZ*q0Q\[oDd^;%
M#^qRIkZEWEgh>[QJJj8Za/j2Nl9iG/"@V9LsG'eanM2mddS=udYY-GCE]"Pnr`5g!Cc-Obbe'u
q"f5D98qZ$j8%fcSS_ns7h_#M.NYZV82!"oE?\\Z,!_Sa:0`5Ta9a2c<Db5'6\bQlD<cd0tbdF%
p@rR:_prRLr!r7Cu$s47P-e^W$ocdqL]f`1[F!Ua^q3ZC1J7ncc]>[_5ZDfg5PJVAuAP*MH+TVS
KdZ*sVB"1u4[^])"Q^:_%bY/g1krrW/D:@:.q4$Z2+:K:S*A8,t'G^Y1!Mia0kSYE$_YHt[J]=u
,"`lZBIcHt"geCE.%aDr!g!"#@5]tD"i]<YWEr;[NeUqFCT^V@S#_8=(,`5T^8a2l?DqTAi]$I$
c?cd0tbdF$CjrR:bqrRLktrR_#!#h@GQdEg(jnbq)>pAY3iM`g\o,<S7]92J\o@:a1lF*N(`Ko(
eOQ'n/7USb&q[Ca&^]">Vh^])"R^:h+dYcl^Xhu<cFKKJT_/NlEj9i4qr@:j:oFaAOjM2d[aR\-
CSXKf.A]"Pnr`Q--Ec-Oece(*%$fVCJ"!!6$+[f3Z9H9qc5!#Hf?\@K2^]=bei^;%Fu_8=+-`5T
a:a2n/"#g1?7bg"GZcd;[=s3gnqrm^nsq:>Pq#h7>NccjVlpAWYCpAY3mY!htE*BQMT92JYm?tF%
iF*E"_Knt\LPaIr3USb&p[Cj3P])oXY^AbnQ^V.7hZ`CY/i;WlX]1q]S/iu9c8P`;g?=R\eF*E"
^Ko1nTR%0hGWiiY8\\#Sl_oBd@bg+S`e'ut"f@YWhpAbC*BWS:t>8I,:)%UDcZa@*J\$rlX]">V
g]t_=t_8=+.`Q#pp`t$56bKJ,ScHab_d*^7hdf._oeG.Soe-OOOda67\al*$Fs7QBmd960(00;B
e8l&Gj?Y!hfEd)k\KSYSKPF%c1U8=fk[Cs>g]=Y_h^;%J!^V7@k[B?LVqr7VXroS#YmQ)d\6:X^
I='T0FCN+B@ItNN8Od;H.U8P&q[(!W[^r"(3b08/XdF6Uqf)F8%DZoe7!s'Z+%Jp,D+a0ibX/rJ
-Yd(O@[C3QS\[oDc]tV:u_SX4g`;ddda2lBFbKS2TcHab_d*^8?df%YleGdqsdKe1Gc-",Ulho6
3p&>*WGWYU[,<nRf:/tG'@q]_!G'\XjLl7=XQCFJ?V5UN&]u%Rt]=bhk^])"S^:_%dZ)Y+nro="
[or:LL3[m-U6qU6T=^GTOD/siIJ;&i?PF%f4Uo:B![CEi__8O@8bK\A\daQatrmq5%D?TG/*<f#
qSY)XOV5C2kX0&P.Z*UgF[^WcX]=bhk^VI\`_@jc)`Q$!@b0.uPc-FV\cd:%ddJVGke,@]$dEp4
abK7i[o`!;=p&>*gW^?D?*^2q^:/b8$@qTUsFa8IgLPq1UQC=A<Uo1<"]Yqe)rk&9F^])"S^V.7
hZ`^1?o&TfPrkhb(3\WWZ6V'sM=Bo9HCN+B@IY3E7OHl9+USb&oZF@BX^r+.4b082YdaQ^rf@SU
'dXDr&!$D@h:P+F`Od)3#S=ZCJUo1/kX0&P/ZEppG\%0)^]Y(uW^Ce8t_8F72`lQ9EbKJ,Sc-FY
]rm:VkrmD5&d*U+bc-+5Lb1Z:_d/We=!rg_Kp,Xlj6V'pM=']9JCiFKAIY*<3NfoZrSY;jVX0B7
Uai(m*]Y2(p^qd_!]XkMUW2@o<jo5DJGWbCT0K_Wj92JSk?XmbdEd)n]Ko(hRQ^j\DW3!51\@]J
k`5g!Cc-Oece(*%$f@A@"XZuO[,TUWqdF>k*M10o*LPq4VQ'e#2T;&-YW2cr$Y->4<[C3TU\[qd
P'"kol^qmq+`Q#s>ai_cLbKS5UrQb;ds3:qqbf\&Ia3j,Grm1W:rrM\4p,X9Y6:X^I='T0GCN"<
>I=[*0NKKKoS=lXSWis%Tc,md#]E,a]rkJiS]XkSYXe_oVroa:_oV"_A3\if^6qL-P=']9ICN+E
AIY3E6OHl6*Tr+imZF7?X^r+.4b08/XdaQ^sf@SR%fBVXV)Yj@p)Md_[s8MrghSYk7G^FspM2m^
_Q^XJ;U84ZbWiW>+Z*LaE[C3NQ\@K2`]t_A!_o'I5a2lBEb0%oNrQP8crlb\lb/q]B`R!Z>rltK
8rrW.o5kd;b4?uA/;-$n0ASQ+(GC4ppM2dX^R%'_CV5^W-`mN,P^:_+m^])"U^V.7h[BQX'[-IY
TrrVr":@:/"4$Q,)9i>%s?tF(kF*E"_Ko1nSQ^j_EW3!51\%KGl`Q-*EcHstfeCE1&e^N.=rVtP
l)Z'L9)KXU/s"aTPo\R0LMh$2)KSb\MPa7`-T:r$WVlHf!Xfnt6ZEgjF[^iu]]Y;1t_Sa=2`Q#s
=aSs<[b5]QiaN)<=_SXU[p\rA9oD\j4:A6e*4$Z5,:fU_-A8,q%GC4poLl@F[R%'_BV5^Q)`RNG
^_7[Fp^V@S"^V.:j[^*!1T!n.fli.(c`a_+!3\WWZ6:X^H<E`^=B5MX2H@CL&N0'<mSYE!]Y-PL
H]thP*aNDcRd*gFof%8L&f'E;6rs@A665g%4)FB54pAZfCnDV9YK7&&sKo(eOPF%].T:r'WVPgD
oXKAY1ZF%'L\[oGe^;%J#_Sa=2`Q#pra8a-c`PfX0^VS.Mo)6]1oD\jUFnRjO5"%t=<*<R<BPha
4H[g['MiX'gR\$4KW2m;Cf%nlt^V.=p^VIY"^:_(f['6O&QFcSPrr`2K>WhF800VZl92APj?=IP
_E-6GSJqf/DPF%c3Uo:B![(!W\_8XL<bg+S`e'ut#f$r77r;?R"m(o-EKpojEo)C6=pZBo4S;;]
GKSPJHOd29%S=Z@HUSXlfX0&S1ZF%'L\[oDc]tV7r^qmkc_@",p^V.:m`7XDNrQ"s0rrW"p55.)
`5!qk:;cm@9BP_[2H@CL%MiNsdR@U"HVlI&;e_o<2_nEat^])"W^:h.h[^*$3S<geZrpTjhrR5p
OmQ*!a5t+FB;cm:5ASQ+(GC+joLl@L^R\$7OWiiY9\\,_q`lZEKcdC1je^i@&e`Zhgs!@[AnD(a
PML^/+Ko(bMP*MB&S=cLNVPpMrXg#(:['mHR\[oDbrk&WP]Y(ke\@9#gg$\dVs7?6lrn(3E3]&u
d7nur`?".G^E-6GSJqf/BOd2?)TVJB`Ye/<AjP%bk]tV4q^qd_!]XkSYYc4@aOh1#Nrr`/IC-M&
H00MTj8kr;d>[V)VDK9oIIY3E6O-H$%TqnZhYd:jO^;@h0b082ZdaQatf@AC2q:tulrV,I"Y`sN
[I"@!.NKBBkR[p(FUnt#iXKJ_1Z*L^Cr3Zj<Za-sN`R3f@rke^+rrVb?557/^4@)G2;-."2Anl7
*G^P'rM2dX_R%0eCV5UN*bi%[De&21!&\l,s^V7Cn\[A`FVk969ZKqJ]rri>ra(dg,3\!3S5t4I
A;cd12A8#h#Fa8FfL5M"TQC=G@VlQu,[^j)e`5]pCcHstfec+,%eD']rgA`Z%r:]C!YF16#JqSo
:NfoWoR@KkAU84ZaWN)u!X/rD*Y-G@GakQ1TqYn;1o)Adi[nQQX,s+=[92JVm?tF(jF*E"^KSYS
KPa@l2Tr"ZgZc(Y`l/'V!^:qA]^^e,m]XkSZYcO[lP),dap\+UiropFc47>3b4?l2&8l&Ae>@1l
RCiFK@I=[*/NKKKoSYDs[Xg,:D]>)5%aNMiSdF6Urf@JI#mJbW8+T;&tf#"\bQ]mSoOcbilQ'[o
.S"HCLVl[5;aO9#4o_c?$nc&XMEVDL>5!qq<<E`^>Bl7s8H@LR&N/s3hS"67KVlR&9fBM\be],A
<^\tqY^:_(g[^*'8U7.0tN3D48qYp`toZM_U47G9d4$>hs7nZWW='K$AB5DO.G^P$pLl@F[R%0h
FW2m//\%98g_oKpDcd:+ie^rF'ceeQGru1n7q!mV/gss`laMGO']t:kocILb5me$>Qrj`!urrW%
u5PR8`4[M\6;cm=8B5MX2H@CI#MiO!dR@KnEVQ-f0d-'lbhp0(;^])"e^V7@l\?rNBVkBE;JTum
Oi:QpEr;#d"P$(!;3\if]6:FI?:fLP$?t3hcDfg2NJ:iZ9O->p"T;&3_Xg5@F]YD>&aihuVdaQa
tf@JI+oZ@.5rXAc'r;-6cq>'g^r;HW#s7$$hjBhUB,s4Ia9N"qs@V0FqFa8CdKo(hPQ'n/7USXo
j[E.=qo'+oL^q[Y`^_ji#]t:kaZ`gC*T9kRjF_YcAOI2>dAk"oP3r_8s5!_V.9M\Sf>$YQJBP_X
/GC+glL5M"SQC=A<V5^Q#[(!Z]_8OC:c-OedeCN4&e_U"Os6BXRrrVnU5P[>d4?uA/;-$n0ASQ.
)G^P$pM2[R]Q^aS@V5UH"^tS/Hn)_m5^VB`_'Yh>o]!o&PY,S4fQ&UB>Bj=M-6U's"s#W&$5X\+
79i+ej>@(cMBl.g2GC+dkL5CnPQ'n/9Uo18sZaREX^r+16bg+VaeCE.%e^s3fJcG$8nG`RhcWU'
)69@A"8l&Gj?Y!kfF*Dt]KSYSJPF%`/TqeNcY.;pCo(2;)b/(j.^qmh$^V.7i\$WB@W2#oLNJ2^
uAQr#'6U*^qpH&=92`s5f6:==:9i+ej>$bTHB52=(FEi.^Jqf,AOd)6'TVJEbY-GCF]>)2$aNDc
SdaQatf@JL8qgne3s6oshn:a%$3[6dT85<)d?=RYcEcubXK85>FP*VN+TVJB`XL6+,nFcD8eA_F
&s1oYi^V7@m\[JiJX/D\^Q&^QHFDPGk<D?%R69md75!1tl5!M:u6:FC99MSGa<`r[7@V'4gDK9l
FI"-d'MN*a`R%'_BVQ-c'[(!W[_8F=9bg4\ce^i@'e_^+Qs60LOrrW/29_g_u4@)G1;HI.4Anu@
-H$t6uMN*a`R%0eCV5UH"^=V`Dp$^h__SEt'_8*k$]t:kc[Bd!9VP0KENJE%/EGB#j=]A3l91_Z
E7Ros>8P2WM:/Oqh='8d8@Us+dD/aQ?H$t3qL5CnPPaIr3U8=ijYdCjL]YMG)aNMlVdaZkJf)jL
UnU^`'s6fmgk]o8d3[QpR78$KX>@;&XDf^,MJ;&f=OHc*$StW!YWirqVjn&3<le'7o_#D.j^q[U
r]=GDXZ)t"%TUV4-LkC(uE,&un>Zk*,;c<Qk2c<RQ<E<4*>@(]IAS5aoDfTuFH[UHuLPq.SPa@l
1TqnWfYHkRG]>)2$aNDcSdF6Xsf@JO<r.4n2s6fmgr4GMp3[HsX8l&Di?Y!kgEd)k[K85DHPEqZ
.TVSHaXL#asnFcMGjjD#^rP9Mi^V.:k\[SrMXf8.iS!K1pKn+JkE,92u@:!5E>Q%b*>;KMi?X[D
SAnc!sDfTrDH@13oKntYJOd2<'StVsXX0/b9\@]Jk`5p*FcdC4kf%8O(iVA')k5XiK!q(+lqDoo
b6:ajM=C,KMD/j]EIY*<3Ng#csS=cRQWN386c04cnp$^b]_Yq=N^`C/&]XkV]Z`pO0V4a?FPDtB
OIX60XDJNs!ARo:\raPn;11Xs/BPVF%E,p&EH$k'kKSG>BO-5crS"?=LVlHo)[C<`\^r477bKeJ
_eCE1&e_9WoJcFm4mJd7dZ:t*U,s+@]9N"qs@:a7nFEi4cKo(eOQ'e&4U8=cgY-c+'n+QMJl.<r
Q_&^>+^:_(h\@&ZHXJhqgSX>Y&MhZh4HZsROE,BE,qe@j$Ci+'/EcZAIH$b!jJq]#<NKB<gR%0e
CV5L>qYd:dK]thM)aNMiSdaQatf@JO<rIP"1s6Taem"mk%3[QpP6qL3T>$kfTDK9rJJ:r]9O->l
uSY;gUWiWG8d,supp[[Fn`;IFl^q[Ur]=PJZZ`gI0VP9ZNQBR>dLP(/+H[0dVF*%5!06;5YGBeC
[I=Hg#L5:_HO-5`oR[g"EV5^MtZ*_!O]thM(aNDcRdF6Urf@JL6qgne.s6Taer5h\03[R!W85<)
c?=RYcEHZ\YK85>FP*VN+TVJB_X08qLg[Oq3p$UY[_u@OQ_&U8*]tCtf\$`NFX/VqiT:;44Oc>3
QKRnW$I!U)8GV=>gH[L6jIt<6,LP^tMOcu&tR[p(GV5UGtYd:dK]YD>%a3)WOd*gFof@SR,o_sF
AjT"QG!r.X:q`6)d5XnFE='T0GCN+EAI=[*/NKKHnS"?@MVlHl(]$'7-p\4=FfuV)1_Z.Fl^q[R
q]=GDXZEL@0Vk]oVR[98$Nf/aLKnFu.JGamLJ:W9'K7nu7Ll..OOcu&sR[ftCUSb#kY-G=B\\,\
n`5g!DcI((he^rC&fBrQgs5X.ErrLGMqDore6qU6T>$kfSDKC#LJ:r]:OHZ$"SY2dUWN<;2^Xqf
Ep\+1?e&VF(s1o\j^V.:l\[SuPYH=h'UnXHNR?j)#O,]'SLkg_>rIt9+s+`%ZM2R:OO-#QjQ^O>
6T;&-ZWiWD0[(!WZ^;7b.aihuVdaQauf@AF8r.4n-s6BUcoU.o?3[[$V85<)d?=RYcEcubXK8,;
EP*VN+T;/6]X0/_:a5?CZp\""4d)Gt#s1oVh^V.:l\[SrOYcb%*VPBfUS!fS,PECodN/[ao!/g`
5s,ACdO-#NgQ'Ri-S=ZFKV5L>qYHbID\\,\n_oBgAc-OedeCN7&e_g1Rs5F"Crr`8TA3BcN8io7
+9Mnhq@:j:oFEi4cKo1kOPa@l2TqnTdXKSqBc/eHhp@R\,bf.K;_SX.*^q[Rq]=GDYZa$^8Wi2b
hTUq[BR$X)&PEM&jr/`@iP*2#nQ'Rf*S"61FUSXlfX0&V3[CEf\^VRk.aihrTdF-Rrf@SR.o7?r
"s69Obq5R'a3\!9[8P`>i?t<thF*Dt]KSYPIPEqZ.TVSHaXKJh:_:IoDp@e(@f?(tK_u@Ll_8*h
!]Xt_a[C!3BXfJG!UnXNRSXZ(9Q^=#(qirRqQ^=,.S"-%@Tq\B\W2cr%Yd:aH]"Gep_oBgAc-Ob
be(*(%e^j-dJcF[.kl1_NKfo8o-p0dc9iG/"@V9LsFaALgLPh+TQ'n,6U8=fgXg#.Ba5$%Qp%7b
6e&VF)s2#Vh^q[Rq]=PM\['Hs>XfJG!V5'cXT:VRCrgWkTrgP4(SXuIHU8+N]W2cr%Yd1XE\@]G
h_8O@8bKS;ZdaZh!f[eX:r.4n*s60Ibrm+V,rAlAh5"&"?<a/sCC2\3=I=[*.N0'6iR[p.JVQ$Z
"Z*h6hi9p40oBb>\`r*^m_SO%&^:_(h\[SuRZEUL5Wi2hmUnaZXTV.qST)YG_Td$5+V5C/hWiW>
,Z*UjI\\#Pj_SjI9b0A5YdF6Xsf[nU+pOWA$s6'C`pS(+S3[m0X85<,e?=R\dEcueZKSPJHP*VN
+TVJB_X0&V3[_L/Jmdp)@khFA%rPSWO(Vmf"]tCqe\@/cNZ*1=4X/`+rVP^2cV#7"dV#R;8Vl?\
sXKJb3Za@0N]"Gbm_SjI9bKS;YdF6Xsf[n['mJ_\:hZ)d="9.>)55dMl4$Q/-;-."3B5DO/H$t9
uMN*a`R%'\@Uo15oYHbIF_U@H3o()2/f$)/5`;daT_A^8+^V.:l]",;X['R$AY-+k,Wi;trW2HP
jVZ*J=W2ZbrX/rG+Yd(L@[^`l[]t_D#`Q-*Dbg+S_e'ut$f[eR3qgne's5s=_rQ\pN3\!6Y8PW8
h?Y*tiF*Dt]K8>JIPEqW-TVJB`XKJe6[CX<)j6Q4'lJU49`VmaT_\9o'_8!^t]Xtbc\$iZMZE^X
;Y-%Z(qlBau+g%oNYd(L?['mHS]=bkn_Sa@5aihrTd*^=mf%8R+eDL(Qs4mY9rrW#,8GtS"4$Q2
.;HI.5B5DR1H@CI#MN*daR%0eCV5L>pYHbID]Z&@Xl0R`thU9QdrP\`Rs2#Db^q[Us]Xtbb\@/f
QZa6sBYd!r,s0*WNZEppF[^WcW]"Gbl_8=.1aND]Ncd:(geCN7(f@ALBJcFL)jSo;QK0]Ar.m--
i:K1M*ASQ+(G^P$pM2[O\Q^XJ<Uo(,mXg#.>\@or2hWF%bin2W"`r<sV`"Bo(_SO%&^:h1k]",>
\[^ENMrj2L-s0Ma4s0_m9)nEPl]Y2(q_SjF6ai_iPcd:+he^i@)f[\U?r.4n&s5X+\flT@U3\3?
W78-T\?".J`EHQSVJqo5DOd2<'St`$YX/rM1ZaREX`6R#njQ#+Jc,n!(`W*mV_[jW#_8*h"]tD"
i]",A]\$u1ArO3iX\[oDb]tV7s_8F42a3)TLcHjkce(*%$f[n^)jo'c1gAg45"8]u948q8m4@)M
6<E`dAC2\0<I"@!-N0'6iR[p+HVQ$W!Yd:^G]"Q&*eD/s<g!\*ca2\(ss2=rT$c9`r^q[Xu]tD"
i]DfD<\c')[]=bei^VI\&_o0O7ai_iPcHsqde(*"#f\"d)h=cF"g&L(3!r8okrAlJn78-T\?".G
`EHZYWJqf2DP*MH*T;&0\Wi`J0['dET]u%k:e^rF%ccaDMrPniUs2=rT#K"<n^qd_"^AbhA]`#M
`^;%J!_Sa:1`lH0CbKS8Xd*^:ke^i@)f[eU-on!.qs5EtZn!?NC3\3?Z85E2g?tF(kFE`+`KSbY
LPa@i0TqeNbXKA_5[C<]Y]tq_3cI((dbf\$*a8X*Y`W*mV_uIUQ_>h=L^\tnG^])"g_8=(,`5Ta
:aND]Lc-FY^dF6Uqf%8R,f%&OErdk+%s5<nYl&%h,3\<N`92Sbq@V9LsFaALhLPh+TQ'n/6U8=f
gXKSn8[^WfZ]thP+b08,SbK@oIrQ"oWrl+oUs24fPpVHmHrkfYm`Q#s>ai_cLbg+P]dF-Lne^rF
*f[eR(n,7h;fDje/0)aaS4ZYVc5Y"RK>$kiTDf^,MJ;&f<OHZ$"SY2aTWN32+ZF.0O]">\l_SsR
;rQ=uXrl>&YrPe]Qr58KOs2FuWs2Yttb0/#Qc-FY^daHXqf%8O+f[eU(l26,4f)OY-0`KR94?>M
d6V:0U>[h;]E-6JUK8,8COd2<'St`$YWiWD/ZaI9Q]=bhl_8F73aN42!r5eiWoZ$pOs2b5^s2ke
obfnAXcd:(fe'uq!fDaD*f%&=7qgndts5*c1qp8q<3]]Gn92\ku@q]_"GC+jmLl7=XQC489U8=f
gXfo%;[^`o\]Y;1t_SjF5a2mtrn]:^OrlY8as31Mg&^ShOdaQ^qe^i@)f[n^(e`Zd[s4%))s"sV
b=?8K)5tFdO>@D,ZE-->RJqf,AOd)3%StVsWWN32,ZaI9Q]"G_k^r""-`Q#s>aN;Qhb5KN`bl>r
fcMu6&dF-IleCE+#f@\a-f$r41qLS[qs4mW1qSu]#3]o`$:f^h1B5DR1H@CL$MiNscR%0eCUo15
oYHbFA\%0,`]t_D#_o0O6a2l?DrQ=NNs31Mf(=(7Pd*U1gdaQ^re^i='f[n^)e'R1PJcF0ug]',
+[U)$V5"/+B=C5TQDK9rKJ;&f<OHZ!!S=lXRW2m&(ZEq!L]">Yi^qmn+`5Ta:aN;TIoZR*Ts3:P
gs3D&!dF$Cje'lgsf%0fO"kM5Qdb4>Fs3^l#s"jGb><=o27SZoe?tF(kFEr:dL5LtQPaIr3TqnT
cXKA_4['mKU]=kqo_Sa=2`lH0BaiaM(qTSl^rQbGhs3LbnrmLhq"4YlOf)=2)f%&9udGXbMs3Uf
!s"jJmBg4[E85<2j@:j:oFaALhL5M"RQ'e&4U8=cfXKJe6[C<]Y]Y2(q_Sa=2`lH0BaiaV+rQP5
brQbAfrm:\mrmLhqs4%)!s475%s4.A'e'ZV%pjrIks4I?,r6KIL4$lM9=']<LD/siIJ:r`:O->l
tS=lXRW2cu'Z*UjI]">Vh^VRe)`5Ta:aN;U(bPo`bc2Q#fci)5idJhSoe,Iksebmu&e^W$odH((
Rs3CYrs"aK/JjVXe92Seu@qfh%G^Y-rLl@FZQ^XJ=USaujXfo%;[CEcZ]Y;.s_Sa=2`lH3Cb5TQ
abl5lecMl/hd/DAkdf._qeGn(ted0gUeC2jle*-[[s3:Sps#g;@O[_K$92\o"ASQ.*H$t6uM2dU
^R%'\@Uo(,lY->4=[^`l[]Y;1t_o'F3`lQ9Eb0.uPbg"DXcHji9d/MGldf%YpeGIere-=CMd*C5
.r.4mls4%'2oWDLa6VC?\?tF+mFa8FfKo1kPPaIr3U84ZcX0/\4['mKU]=bkm_8=+.`Q-'Ab0%o
Nbfn>WcHjh`d/DAldf%YoeGe"te-FINd*Btkp&9OBbQ$&k4S?O$6:=LG>%)#ZEHQSVJqo5DP*ME
(T:r'YWN<8,ZF.-N]">Vh^qmq+`5Ta;ai_cLbKS2TcHab^d*Vd>rmLSjs3_/"d*Kq\iV!2$s2tA
is#g88ND2c4;d*R@CiOTDIY<K7NfoZpS=ZFMVQ-]"Yd1XE\@T;b^;%M$_o0O7a2uKHbKJ,Rc-=P
[cd:&<dJ_Mke,Ri$dEp1_aNN]CJcEgkdJl9,m&OD]8lJo$AnuC0H@LR&MiX$dR@BhCUo12nY->4
=[^`l[]Y;.r_Sa=2`lH0Bb0.uPbg"DXcd2U;rR(YnrQtntcd'bVajKANJcEdjci5rua+-`Q;HdI
?CiOTEItNN7O-5frS=cOOVQ-`#Yd1UC\%0,`]t_A"_SjF4a2lBEb0.uPbg"E3cMc)hd/MAkc3MV
<b/hc_pATXC`rF?a2Z<*KD,3u0?Y+"lFa8FgL5V(TQ'e&4TqeKaX0&V2ZaI9Q]">Yi^qmn*`5Ta
:aN2KGbKJ,Rrlt;b$-g];b/hTFg@P5ns2=r]s"X?&P$_!b?"IbiFEr=eL5CqQPaIr3TVJ?^WiWA
.ZF%'M]">Vg^;7Y&_o0O6a2c9CrlY5`r6,Jjb0%iH`Pg'cq18R_s3(Etrq=d$?W:6<BPhg8I"?s
-NKB?iR@KnDV5L>pY-5+:[C<ZV]=bkm^qmn*`5T^8a2l?uaoBEfa2Z*9_p7<8rdk*cs2k9prU7U
V@p<>QCiOWFJ:r]9O->ltS"?=LVQ$VuY->4=[C<]X]=bhk^qmn*_o0L4`VdaU`<aE"_SEt0ht6k
us1eTSs"46Bd\(G`@:sFuG^Y-sM2[O[QC=>9U84]dX0&P/Z*UjH\@K2_]Y2%o^qde&r5&cV^q[U
r]Z/h$rdk*_s2G!frU\=.G%>,uDg$GVKSPJGP*ME(StMgSVl?`!Y->4<['mHR\[oDb]Y2#W^'VK
c]XkYdakcI`JcE@^_uD%^qW5>QGAD#6G'e^lLPq4VQ'e#2T:r'XW2co#Y->1:Za@-K\,<c9[g0=
I[_^ASq#5jE\c9JF+8k]`]:FUqEcl\WK85>DOd)3#S=ZCJUo()iX/rG+YPtd)Z2_$4Zan04nb`<
:s0r$Aru1h,hn,J9H?OO]JVAr>O-5`oR@KkAU8+N]VZ*LoWW&h$X0][hhsL4;JcE+W])O#RqX`a
m[@rnINJrdQN/`m\Pa7Z'R[p+HW2m)2a4KJGqu-MFs0;U4ru1n5oBG/cdDr])YcP(7ZEUI?_TC<
ck3DO6rIP!Ms0)G3rqu]jq>'m`q=sgarVhBJVZ25IJcEaiJcC<$`rCViJcEaiJcC<$`rCViJcEa
iJcC<$`rCP~>
%%EndData
showpage
%%Trailer
end
%%EOF
---------------0907121236356--
