KöMaL 2016 B4826

The circles $a$ and $b$ intersect at two distinct points $P$ and $Q$. Suppose that $A_0$ is a point that lies on circle $a$. Define the points $B_0$, $A_1$, $B_1$, $A_2$ etc., respectively, as follows: $B_i$ is the intersection of line $A_iP$ with $b$, and then $A_{i+1}$ is the intersection of line $B_iQ$ with $a$. (If $A_i = P$ then line $A_iP$ means the tangent of circle $a$ at $P$, and if $B_i = Q$ then line $B_iQ$ means the tangent of circle $b$ at $Q$.) Show that if there exists an $A_n$ ($n\ge 2$) for which $A_0$ and $A_n$ coincide, then this holds for any point $A_0$ of circle $a$.