\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} \newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} \newcommand{\set}[1]{\left\{ #1 \right\}} \newcommand{\norm}[1]{\left\| #1 \right\|} \newcommand{\rm}[1]{\operatorname{#1}} \newcommand{\F}{\mathbb{F}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\e}{\varepsilon} \newcommand{\mex}{\operatorname{mex}} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\poly}{\operatorname{poly}} \newcommand{\polylog}{\operatorname{polylog}} \newcommand{\span}{\operatorname{span}}