MK-MO 2019 Problem 4

Determine all functions $f: \mathbb {N} \to \mathbb {N}$ such that $n!\hspace{1mm} +\hspace{1mm} f(m)!\hspace{1mm} |\hspace{1mm} f(n)!\hspace{1mm} +\hspace{1mm} f(m!)$ ,

for all $m$ , $n$ $\in$ $\mathbb{N}$ .