CMO 2015 Problem 1

Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^{2}<f(n) f(f(n))<n^{2}+n$ for every positive integer $n$.