KMO 2023 Problem 2

Function $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies the following condition.

(Condition) For each positive real number $x$ , there exists a positive real number $y$ such that $(x + f(y))(y + f(x)) \leq 4$ , and the number of $y$ is finite.

Prove $f(x) > f(y)$ for any positive real numbers $x < y$ . ( $\mathbb{R^+}$ is a set for all positive real numbers.)