EGMO 2022 Problem 2

Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold: (1) $f(a b)=f(a) f(b)$, and (2) at least two of the numbers $f(a), f(b)$ and $f(a+b)$ are equal.

Proposed by: Fedir Yudin, Ukraine