MEMO 2025 T2

Let $\mathbb{R}^+$ be the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all numbers $x, y \in \mathbb{R}^+$, we have

$$f(xy) + f(x) = f(y)f(xf(y)) + f(x)f(y),$$

and there exists at most one number $a \in \mathbb{R}^+$ such that $f(a) = 1.$