$$
\def\ba{\mathbf{a}}
\def\bb{\mathbf{b}}
\def\bc{\mathbf{c}}
\def\bd{\mathbf{d}}
\def\be{\mathbf{e}}
\def\bf{\mathbf{f}}
\def\bg{\mathbf{g}}
\def\bh{\mathbf{h}}
\def\bi{\mathbf{i}}
\def\bj{\mathbf{j}}
\def\bk{\mathbf{k}}
\def\bl{\mathbf{l}}
\def\bm{\mathbf{m}}
\def\bn{\mathbf{n}}
\def\bo{\mathbf{o}}
\def\bp{\mathbf{p}}
\def\bq{\mathbf{q}}
\def\br{\mathbf{r}}
\def\bs{\mathbf{s}}
\def\bt{\mathbf{t}}
\def\bu{\mathbf{u}}
\def\bv{\mathbf{v}}
\def\bw{\mathbf{w}}
\def\bx{\mathbf{x}}
\def\by{\mathbf{y}}
\def\bz{\mathbf{z}}
\def\bA{\mathbf{A}}
\def\bB{\mathbf{B}}
\def\bC{\mathbf{C}}
\def\bD{\mathbf{D}}
\def\bE{\mathbf{E}}
\def\bF{\mathbf{F}}
\def\bG{\mathbf{G}}
\def\bH{\mathbf{H}}
\def\bI{\mathbf{I}}
\def\bJ{\mathbf{J}}
\def\bK{\mathbf{K}}
\def\bL{\mathbf{L}}
\def\bM{\mathbf{M}}
\def\bN{\mathbf{N}}
\def\bO{\mathbf{O}}
\def\bP{\mathbf{P}}
\def\bQ{\mathbf{Q}}
\def\bR{\mathbf{R}}
\def\bS{\mathbf{S}}
\def\bT{\mathbf{T}}
\def\bU{\mathbf{U}}
\def\bV{\mathbf{V}}
\def\bW{\mathbf{W}}
\def\bX{\mathbf{X}}
\def\bY{\mathbf{Y}}
\def\bZ{\mathbf{Z}}
\def\balpha{\boldsymbol{\alpha}}
\def\bbeta{\boldsymbol{\beta}}
\def\bgamma{\boldsymbol{\gamma}}
\def\bdelta{\boldsymbol{\delta}}
\def\bepsilon{\boldsymbol{\epsilon}}
\def\bvarepsilon{\boldsymbol{\varepsilon}}
\def\bzeta{\boldsymbol{\zeta}}
\def\btheta{\boldsymbol{\theta}}
\def\bvartheta{\boldsymbol{\vartheta}}
\def\biota{\boldsymbol{\iota}}
\def\bkappa{\boldsymbol{\kappa}}
\def\blambda{\boldsymbol{\lambda}}
\def\bmu{\boldsymbol{\mu}}
\def\bnu{\boldsymbol{\nu}}
\def\bxi{\boldsymbol{\xi}}
\def\bpi{\boldsymbol{\pi}}
\def\brho{\boldsymbol{\rho}}
\def\bsigma{\boldsymbol{\sigma}}
\def\bvarsigma{\boldsymbol{\varsigma}}
\def\btau{\boldsymbol{\tau}}
\def\bupsilon{\boldsymbol{\upsilon}}
\def\bphi{\boldsymbol{\phi}}
\def\bvarphi{\boldsymbol{\varphi}}
\def\bchi{\boldsymbol{\chi}}
\def\bpsi{\boldsymbol{\psi}}
\def\bomega{\boldsymbol{\omega}}
\def\bGamma{\boldsymbol{\Gamma}}
\def\bDelta{\boldsymbol{\Delta}}
\def\bTheta{\boldsymbol{\Theta}}
\def\bLambda{\boldsymbol{\Lambda}}
\def\bXi{\boldsymbol{\Xi}}
\def\bPi{\boldsymbol{\Pi}}
\def\bSigma{\boldsymbol{\Sigma}}
\def\bUpsilon{\boldsymbol{\Upsilon}}
\def\bPhi{\boldsymbol{\Phi}}
\def\bPsi{\boldsymbol{\Psi}}
\def\bOmega{\boldsymbol{\Omega}}
\def\bba{\mathbb{a}}
\def\bbb{\mathbb{b}}
\def\bbc{\mathbb{c}}
\def\bbd{\mathbb{d}}
\def\bbe{\mathbb{e}}
\def\bbf{\mathbb{f}}
\def\bbg{\mathbb{g}}
\def\bbh{\mathbb{h}}
\def\bbi{\mathbb{i}}
\def\bbj{\mathbb{j}}
\def\bbk{\mathbb{k}}
\def\bbl{\mathbb{l}}
\def\bbm{\mathbb{m}}
\def\bbn{\mathbb{n}}
\def\bbo{\mathbb{o}}
\def\bbp{\mathbb{p}}
\def\bbq{\mathbb{q}}
\def\bbr{\mathbb{r}}
\def\bbs{\mathbb{s}}
\def\bbt{\mathbb{t}}
\def\bbu{\mathbb{u}}
\def\bbv{\mathbb{v}}
\def\bbw{\mathbb{w}}
\def\bbx{\mathbb{x}}
\def\bby{\mathbb{y}}
\def\bbz{\mathbb{z}}
\def\bbA{\mathbb{A}}
\def\bbB{\mathbb{B}}
\def\bbC{\mathbb{C}}
\def\bbD{\mathbb{D}}
\def\bbE{\mathbb{E}}
\def\bbF{\mathbb{F}}
\def\bbG{\mathbb{G}}
\def\bbH{\mathbb{H}}
\def\bbI{\mathbb{I}}
\def\bbJ{\mathbb{J}}
\def\bbK{\mathbb{K}}
\def\bbL{\mathbb{L}}
\def\bbM{\mathbb{M}}
\def\bbN{\mathbb{N}}
\def\bbO{\mathbb{O}}
\def\bbP{\mathbb{P}}
\def\bbQ{\mathbb{Q}}
\def\bbR{\mathbb{R}}
\def\bbS{\mathbb{S}}
\def\bbT{\mathbb{T}}
\def\bbU{\mathbb{U}}
\def\bbV{\mathbb{V}}
\def\bbW{\mathbb{W}}
\def\bbX{\mathbb{X}}
\def\bbY{\mathbb{Y}}
\def\bbZ{\mathbb{Z}}
\def\cA{\mathcal{A}}
\def\cB{\mathcal{B}}
\def\cC{\mathcal{C}}
\def\cD{\mathcal{D}}
\def\cE{\mathcal{E}}
\def\cF{\mathcal{F}}
\def\cG{\mathcal{G}}
\def\cH{\mathcal{H}}
\def\cI{\mathcal{I}}
\def\cJ{\mathcal{J}}
\def\cK{\mathcal{K}}
\def\cL{\mathcal{L}}
\def\cM{\mathcal{M}}
\def\cN{\mathcal{N}}
\def\cO{\mathcal{O}}
\def\cP{\mathcal{P}}
\def\cQ{\mathcal{Q}}
\def\cR{\mathcal{R}}
\def\cS{\mathcal{S}}
\def\cT{\mathcal{T}}
\def\cU{\mathcal{U}}
\def\cV{\mathcal{V}}
\def\cW{\mathcal{W}}
\def\cX{\mathcal{X}}
\def\cY{\mathcal{Y}}
\def\cZ{\mathcal{Z}}
\def\ta{\textnormal{a}}
\def\tb{\textnormal{b}}
\def\tc{\textnormal{c}}
\def\td{\textnormal{d}}
\def\te{\textnormal{e}}
\def\tf{\textnormal{f}}
\def\tg{\textnormal{g}}
\def\th{\textnormal{h}}
\def\ti{\textnormal{i}}
\def\tj{\textnormal{j}}
\def\tk{\textnormal{k}}
\def\tl{\textnormal{l}}
\def\tm{\textnormal{m}}
\def\tn{\textnormal{n}}
\def\to{\textnormal{o}}
\def\tp{\textnormal{p}}
\def\tq{\textnormal{q}}
\def\tr{\textnormal{r}}
\def\ts{\textnormal{s}}
\def\tt{\textnormal{t}}
\def\tu{\textnormal{u}}
\def\tv{\textnormal{v}}
\def\tw{\textnormal{w}}
\def\tx{\textnormal{x}}
\def\ty{\textnormal{y}}
\def\tz{\textnormal{z}}
\def\tA{\textnormal{A}}
\def\tB{\textnormal{B}}
\def\tC{\textnormal{C}}
\def\tD{\textnormal{D}}
\def\tE{\textnormal{E}}
\def\tF{\textnormal{F}}
\def\tG{\textnormal{G}}
\def\tH{\textnormal{H}}
\def\tI{\textnormal{I}}
\def\tJ{\textnormal{J}}
\def\tK{\textnormal{K}}
\def\tL{\textnormal{L}}
\def\tM{\textnormal{M}}
\def\tN{\textnormal{N}}
\def\tO{\textnormal{O}}
\def\tP{\textnormal{P}}
\def\tQ{\textnormal{Q}}
\def\tR{\textnormal{R}}
\def\tS{\textnormal{S}}
\def\tT{\textnormal{T}}
\def\tU{\textnormal{U}}
\def\tV{\textnormal{V}}
\def\tW{\textnormal{W}}
\def\tX{\textnormal{X}}
\def\tY{\textnormal{Y}}
\def\tZ{\textnormal{Z}}
\newcommand{\defeq}{\overset{\text{def}}{=}}
\newcommand{\smallR}{\mbox{\tiny R}}
\newcommand{\smallF}{\mbox{\tiny F}}
\newcommand{\softmax}{\mathrm{softmax}}
\newcommand{\sigmoid}{\sigma}
\newcommand{\KL}{D_{\mathrm{KL}}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Cov}{\mathrm{Cov}}
$$
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