KZ-MO 2022 Problem 3

Given $m\in\mathbb{N}$ . Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$ holds for all $x,y\in\mathbb{R^{+}}.$ ( $f^m(x) =$ $f$ applies $m$ times.)