APMO 2023 Problem 4

Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that

$$f((c+1) x+f(y))=f(x+2 y)+2 c x \text { for all } x, y \in \mathbb{R}_{>0}$$