SHARYGIN 2021 Problem 84

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$ . Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$ . Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$ .