2023 AIME II Problem 12

In \triangle ABC with side lengths AB = 13, BC = 14, and CA = 15, let M be the midpoint of \overline{BC}. Let P be the point on the circumcircle of \triangle ABC such that M is on \overline{AP}. There exists a unique point Q on segment \overline{AM} such that \angle PBQ = \angle PCQ. Then AQ can be written as \frac{m}{\sqrt{n}}, where m and n are relatively prime positive integers. 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.