SHARYGIN 2009 Problem 4

Let $P$ and $Q$ be the common points of two circles. The ray with origin $Q$ reflects from the first circle in points $A_1, A_2, \ldots$ according to the rule “the angle of incidence is equal to the angle of reflection”. Another ray with origin $Q$ reflects from the second circle in the points $B_1, B_2, \ldots$ in the same manner. Points $A_1, B_1$ and $P$ occurred to be collinear. Prove that all lines $A_iB_i$ pass through $P$.