SHARYGIN 2020 Problem 8

Two circles meeting at points $P$ and $R$ are given. Let $\ell_1$ , $\ell_2$ be two lines passing through $P$ . The line $\ell_1$ meets the circles for the second time at points $A_1$ and $B_1$ . The tangents at these points to the circumcircle of triangle $A_1RB_1$ meet at point $C_1$ . The line $C_1R$ meets $A_1B_1$ at point $D_1$ . Points $A_2$ , $B_2, C_2, D_2$ are defined similarly. Prove that the circles $D_1D_2P$ and $C_1C_2R$ touch.