Prove that there exists a function $f : \mathbb{N} \mapsto \mathbb{N}$ that satisfies the following: ___1. For all positive integers $m, n$ we have \[\gcd(|f(m)-f(n)|, f(mn)) = f(\gcd(m, n))\]___2. For all positive integers $m$, we have $f(f(m)) = f(m)$. ___3. For all positive integers $k$, there exists a positive integer $n$ with $2024^{k} \mid f(n)$. Proposed by MV Adhitya, Archit Manas