BALTIC-WAY 2022 Problem 16

Let $\mathbb{Z^+}$ denote the set of positive integers. Find all functions $f:\mathbb{Z^+} \to \mathbb{Z^+}$ satisfying the condition $$f(a) + f(b) \mid (a + b)^2$$ for all $a,b \in \mathbb{Z^+}$