\( \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\id}{\mathrm{id}} \DeclareMathOperator{\re}{Re} \DeclareMathOperator{\im}{Im} \DeclareMathOperator{\syt}{syt} \DeclareMathOperator{\vir}{span} \DeclareMathOperator{\image}{Im} \)