\( \newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \definecolor{orange}{RGB}{255, 165, 0} \definecolor{crimson}{RGB}{220,20,60} \definecolor{pink}{RGB}{255,20,147} \definecolor{darkGreen}{RGB}{34,139,34} \definecolor{olive}{RGB}{128,128,0} \definecolor{teal}{RGB}{0,128,128} \definecolor{turquoise}{RGB}{64,224,208} \definecolor{navy}{RGB}{30, 144, 255} \definecolor{purple}{RGB}{128,0,128} \definecolor{indigo}{RGB}{216, 191, 216} \definecolor{maroon}{RGB}{128,0,0} \definecolor{gold}{RGB}{255,215,0} \definecolor{lemon}{RGB}{255, 250, 205} \definecolor{peru}{RGB}{205, 133, 63} \definecolor{alizarin}{rgb}{0.82, 0.1, 0.26} \definecolor{brightmaroon}{rgb}{0.76, 0.13, 0.28} \definecolor{cardinal}{rgb}{0.77, 0.12, 0.23} \definecolor{tuscanred}{rgb}{0.51, 0.21, 0.21} \definecolor{steelBlue}{rgb}{0.27, 0.51, 0.71} \definecolor{tyrianpurple}{rgb}{0.4, 0.01, 0.24} \def\Z{\mathbb Z} \def\R{\mathbb R} \def\Q{\mathbb Q} \def\N{\mathbb N} \def\C{\mathbb C} \def\ba{{\mathbf{a}}} \def\bb{\mathbf{b}} \def\bh{{\mathbf{h}}} \def\bu{{\mathbf{u}}} \def\bv{{\mathbf{v}}} \def\bw{{\mathbf{w}}} \def\bx{{\mathbf{x}}} \def\by{\mathbf{y}} \def\bone{{\mathbf{1}}} \def\bzero{\mathbf{0}} \def\var{{\mbox{var}}} \def\P{{\mathbb{P}}} \def\E{{\mathbb{E}}} \def\cA{{\mathcal{A}}} \def\cB{{\mathcal{B}}} \def\cC{{\mathcal{C}}} \def\cP{\mathcal P} \def\cF{{\mathcal F}} \def\cE{\mathcal E} \def\cO{\mathcal O} \def\kin{k^{\mathrm{in}}} \def\kout{k^{\mathrm{out}}} \newcommand{\indeg}[1]{k^{\mathrm{in}}_{#1}} \newcommand{\outdeg}[1]{k^{\mathrm{out}}_{#1}} \tikzset{->-/.style={decoration={ markings, mark=at position .5 with {\arrow{latex}}},postaction={decorate}}} \newcommand{\pile}[1]{\bullet{#1}} \newcommand{\ppile}[1]{\bullet({#1})} \newcommand{\nim}[1]{*{#1}} \newcommand{\bc}{C} \newcommand{\R}{{\mathbb R}} \def\nimsum{\, \oplus \,} \def\mex{{\mathrm{mex}}} \def\g{g} \newcommand{\inter}{{\mathrm{int}}} \def\cA{{\mathcal A}} \def\cB{{\mathcal B}} \def\cF{{\mathcal F}} \def\cG{{\mathcal G}} \def\cH{{\mathcal H}} \def\cN{{\mathcal N}} \def\cQ{{\mathcal Q}} \def\cP{{\mathcal P}} \def\W{{\mathbb W}} \newcommand{\nimbleboard}[1]{ \begin{scope}[shift={(.25, .4)}] \draw (0,-.3) -- (#1,-.3) -- (#1+.45, .7) -- (#1+.45, 1); \draw (0,0) -- (#1,0); \draw (.45,1) -- (#1+.45,1); \foreach \i in {0, ..., #1} { \draw (\i, -.3) -- (\i, 0) -- (\i+.45,1); } \foreach \i in {1, ..., #1} { \node[style={font=\sffamily\scriptsize}] at (\i - .5, -.15) {\i}; } \end{scope} } \newcommand{\coin}[1]{ \begin{scope}[shift={#1}] \draw[color=white, fill=white] (-.4, 0) -- (-.4, -.1) -- (.4,-.1) -- (.4,0) -- cycle; \draw[thick, fill=white] (0,0) ellipse (0.4 and 0.10); \draw[thick, fill=white] (.4,-.1) arc (0:-180:0.4 and 0.10); \draw[thick] (0.4,0) -- (0.4, -.1); \draw[thick] (-0.4,0) -- (-0.4, -.1); \end{scope} } \newcommand{\bean}[2]{ \begin{scope}[shift={#1}, rotate={#2}] \draw [very thick, fill=gray!50] plot [smooth cycle] coordinates {(-.2,0) (-.175, .07) (-.1,.1) (0,.075) (.1,.1) (.175, .07) (.2,0) (.15, -.07) (0,-.1) (-.15, -.07)}; \end{scope} } \newcommand{\heap}[2]{ \begin{scope}[shift={#1}] \draw[very thick] (0,0) circle ({#2}); \end{scope} } \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)