Limit, Continuity and Differentiability: JEE Main 2023 Mathematics question

Let $[x]$ denote the greatest integer function and $f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2$. 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 :