Mathematics Question from JEE Main 2023 · 15 April, Shift 1

Let $[x]$ denote the greatest integer function and $f(x)=\max\{1+x+[x],\,2+x,\,x+2[x]\},\quad 0\le x\le 2.$ Let $m$ be the number of points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in $(0,2)$, where $f$ is not differentiable. Then $(m+n)^2+2$ is equal to