Meromorphic function that preserves the upper half-plane and the lower half plane must have only simple poles, all of them in $\mathbb{R}$

Question

Let $f: \mathbb{C} \to \mathbb{C} $ be a meromorphic non-constant function such that preserves the upper half-plane and the lower half plane. Show that all poles of $f$ are simple and lie in $\mathbb{R}$.

By continuity, I've already noticed that $f$ must preserve $\mathbb{R}$ but I don't know if this is useful at all or what else to do, so I would appreciate any hint. Thank you!