US-TST 2007 Problem 1

Circles $\omega_1$ and $\omega_2$ meet at $P$ and $Q$ . Segments $AC$ and $BD$ are chords of $\omega_1$ and $\omega_2$ respectively, such that segment $AB$ and ray $CD$ meet at $P$ . Ray $BD$ and segment $AC$ meet at $X$ . Point $Y$ lies on $\omega_1$ such that $PY \parallel BD$ . Point $Z$ lies on $\omega_2$ such that $PZ \parallel AC$ . Prove that points $Q,X,Y,Z$ are collinear.