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

Let's think step by step and output the final answer within \boxed{}.
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