SHARYGIN 2014 Problem 12

Circles $\omega_1$ and $\omega_2$ meet at points $A$ and $B$ . Let points $K_1$ and $K_2$ of $\omega_1$ and $\omega_2$ respectively be such that $K_1A$ touches $\omega_2$ , and $K_2A$ touches $\omega_1$ . The circumcircle of triangle $K_1BK_2$ meets lines $AK_1$ and $AK_2$ for the second time at points $L_1$ and $L_2$ respectively. Prove that $L_1$ and $L_2$ are equidistant from line $AB$ .