SHARYGIN 2022 Problem 10

Let $\omega_1$ be the circumcircle of triangle $ABC$ and $O$ be its circumcenter. A circle $\omega_2$ touches the sides $AB, AC$ , and touches the arc $BC$ of $\omega_1$ at point $K$ . Let $I$ be the incenter of $ABC$ . Prove that the line $OI$ contains the symmedian of triangle $AIK$ .