Let \overline{A B} be a chord of a circle \omega, and let P be a point on the chord \overline{A B}. Circle \omega_{1} passes through A and P and is internally tangent to \omega. Circle \omega_{2} passes through B and P and is internally tangent to \omega. Circles \omega_{1} and \omega_{2} intersect at points P and Q. Line P Q intersects \omega at X and Y. Assume that A P=5, P B=3, X Y=11, and P Q^{2}=\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.