2023 AIME II Problem 12

In \triangle A B C with side lengths A B=13, B C=14, and C A=15, let M be the midpoint of \overline{B C}. Let P be the point on the circumcircle of \triangle A B C such that M is on \overline{A P}. There exists a unique point Q on segment \overline{A M} such that \angle P B Q=\angle P C Q. Then A Q can be written as \frac{m}{\sqrt{n}}, where m and n are relatively prime positive integers. Find m+n.