RU-ALLRUSSIAN 2022 Problem 98

A circle $\omega$ is inscribed in triangle $ABC$ , tangent to the side $BC$ at point $K$ . Circle $\omega'$ is symmetrical to the circle $\omega$ with respect to point $A$ . The point $A_0$ is chosen so that the segments $BA_0$ and $CA_0$ touch $\omega'$ . Let $M$ be the midpoint of side $BC$ . Prove that the line $AM$ bisects the segment $KA_0$ .