Middle European Mathematical Olympiad 2022 Problem T-7

Let $\mathbb{N}$ denote the set of positive integers. Determine all functions $f\colon \mathbb{N}\to \mathbb{N}$ such that $f(1)\leq f(2)\leq f(3)\leq \ldots$ and the numbers $f(n) + n + 1$ and $f(f(n)) - f(n)$ are both perfect squares for every positive integer $n$.