2023 AIME II Problem 9

Circles \omega_{1} and \omega_{2} intersect at two points P and Q, and their common tangent line closer to P intersects \omega_{1} and \omega_{2} at points A and B, respectively. The line parallel to A B that passes through P intersects \omega_{1} and \omega_{2} for the second time at points X and Y, respectively. Suppose P X=10, P Y=14, and P Q=5. Then the area of trapezoid X A B Y is m \sqrt{n}, where m and n are positive integers and n is not divisible by the square of any prime. Find m+n.