functional equations - If $f(x) = f(y)$ or $f(frac{x+y}{2}) = f(sqrt{xy})$, find $f$

Assume $f: (0, \infty) \to \mathbb{R}$ is a continuous function such that for any $x,y > 0$, $f(x) = f(y)$ or $f(\frac{x+y}{2}) = f(\sqrt{xy})$. Find $f$.

I would work with each condition separately then combine later. So for $f(\frac{x+y}{2}) = f(\sqrt{xy})$ we see that the argument of each side is part of AM-GM, so that might help. I am wondering though how I might derive anything from the second equation since all I can see is that $f(\frac{x+1}{2}) = f(\sqrt{x})$, which I don't see how helps.