$$ABCD$$ is a Cyclic Quadrilateral, with $$\displaystyle \overline{AB}=11$$, $$\displaystyle \overline{CD}=19$$. $$P$$ and $$Q$$ are points on $$\overline{AB}$$ and $$\overline{CD}$$ such that $$\displaystyle \overline{AP}=6$$ and $$\displaystyle \overline{DQ}=7$$. We have $$\displaystyle \overline{PQ}=27.$$ Determine the length of the line segment formed when $$\displaystyle \overline{PQ}$$ is extended from both sides until it reaches the circle.