BALTIC-WAY 2025 Problem 18

Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(2) = 1$ and

$$\text{lcm}(f(a+b), f(b)) \mid \text{lcm}(a+f(b), b)$$

for all positive integers $a, b$.