BALKAN-SL 2021 G2

Let $I$ and $O$ be the incenter and the circumcenter of a triangle $A B C$, respectively, and let $s_{a}$ be the exterior bisector of angle $\angle B A C$. The line through $I$ perpendicular to $I O$ meets the lines $B C$ and $s_{a}$ at points $P$ and $Q$, respectively. Prove that $I Q=2 I P$.