\( \newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \def\Z{\mathbb Z} \def\R{\mathbb R} \def\Q{\mathbb Q} \def\N{{\mathbb N}} \def\C{\mathbb C} \def\ba{{\mathbf{a}}} \def\bb{\mathbf{b}} \def\bh{{\mathbf{h}}} \def\bu{{\mathbf{u}}} \def\bv{{\mathbf{v}}} \def\bw{{\mathbf{w}}} \def\bx{{\mathbf{x}}} \def\by{\mathbf{y}} \def\bone{{\mathbf{1}}} \def\bzero{\mathbf{0}} \def\var{{\mbox{var}}} \def\P{{\mathbb{P}}} \def\E{{\mathbb{E}}} \def\cA{{\mathcal{A}}} \def\cB{{\mathcal{B}}} \def\cC{{\mathcal{C}}} \def\cP{\mathcal P} \def\cE{\mathcal E} \def\cO{\mathcal O} \def\kin{k^{\mathrm{in}}} \def\kout{k^{\mathrm{out}}} \newcommand{\indeg}[1]{k^{\mathrm{in}}_{#1}} \newcommand{\outdeg}[1]{k^{\mathrm{out}}_{#1}} \tikzset{->-/.style={decoration={ markings, mark=at position .5 with {\arrow{latex}}},postaction={decorate}}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)