SHARYGIN 2009 Problem 12

Let $CL$ be a bisector of triangle $ABC$ . Points $A_1$ and $B_1$ are the reflections of $A$ and $B$ in $CL$ , points $A_2$ and $B_2$ are the reflections of $A$ and $B$ in $L$ . Let $O_1$ and $O_2$ be the circumcenters of triangles $AB_1B_2$ and $BA_1A_2$ respectively. Prove that angles $O_1CA$ and $O_2CB$ are equal.