MathNet 2024-part-03 — 168 problems (EN–VI) Question 48

For a function $f : \mathbb{N} \to \mathbb{N}$ define $P(n) = f(1) \cdots f(n)$. Find all functions $f$ such that for all positive integers $a$ and $b$, we have
$$
P(a) + P(b) \mid a! + b!
$$