Nemotron-IMO-Bench 2026 A33

Let $\mathbf{Z}_{>0}$ be the set of positive integers. Find all functions $f:\mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(m)\geq m$ and $$f(m+n) \mid f(m)+f(n)$$ for all $m,n\in \mathbb{Z}_{>0}$.