CH-SMO 2021 Problem 6

Let $\mathbb{N}$ be the set of positive integers. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that for every positive integer $n \in \mathbb{N}$ $$f(n) -n<2021 \text{and} f^{f(n)}(n) =n$$ Prove that $f(n)=n$ for infinitely many $n \in \mathbb{N}$