2020 IMO Shortlist A8

Let $\mathbb{R}^{+}$be the set of positive real numbers. Determine all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that, for all positive real numbers $x$ and $y$, $$f(x+f(x y))+y=f(x) f(y)+1$$

令 $\mathbb{R}^{+}$ 为正实数集合。确定所有函数 $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$，使得对于所有正实数 $x$ 和 $y$，$$f(x+f(x y))+y=f(x) f(y)+1$$