HMMT 2006 G3

Let $A, B, C$, and $D$ be points on a circle such that $A B=11$ and $C D=19$. Point $P$ is on segment $A B$ with $A P=6$, and $Q$ is on segment $C D$ with $C Q=7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $P Q=27$, find $X Y$.