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 \overline{AB} that passes through P intersects \omega_1 and \omega_2 for the second time at points X and Y, respectively. Suppose PX = 10, PY = 14, and PQ = 5. Then the area of trapezoid XABY 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.