FKMO 2010 Problem 4

Given is a trapezoid $ABCD$ where $AB$ and $CD$ are parallel, and $A,B,C,D$ are clockwise in this order. Let $\Gamma_1$ be the circle with center $A$ passing through $B$ , $\Gamma_2$ be the circle with center $C$ passing through $D$ . The intersection of line $BD$ and $\Gamma_1$ is $P$ $( \ne B,D)$ . Denote by $\Gamma$ the circle with diameter $PD$ , and let $\Gamma$ and $\Gamma_1$ meet at $X$ $( \ne P)$ . $\Gamma$ and $\Gamma_2$ meet at $Y$ . If the circumcircle of triangle $XBY$ and $\Gamma_2$ meet at $Q$ , prove that $B,D,Q$ are collinear.