2018 AIME II Problem 14

The incircle \omega of triangle ABC is tangent to \overline{BC} at X. Let Y \neq X be the other intersection of \overline{AX} with \omega. Points P and Q lie on \overline{AB} and \overline{AC}, respectively, so that \overline{PQ} is tangent to \omega at Y. Assume that AP = 3, PB = 4, AC = 8, and AQ = \dfrac{m}{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.