Let $\mathbb{R}_{+}$ be the set of all positive real numbers. Find all functions $f{:}~ \mathbb{R}_{+} \mapsto \mathbb{R}_{+}$ such that for all $x,y\in R_{+}$, we have $$ f(x)=f(f(f(x))+y)+f(xf(y))f(x+y). $$
This is the third question from USAMO 2022. I tried to solve it but failed because of the annoying $f(f(x))$. However, I couldn’t find an official answer since it is written “Work In Process”.