The incircle \omega of \triangle A B C is tangent to \overline{B C} at X. Let Y \neq X be the other intersection of \overline{A X} and \omega. Points P and Q lie on \overline{A B} and \overline{A C}, respectively, so that \overline{P Q} is tangent to \omega at Y. Assume that A P=3, P B=4, A C=8, and A Q=\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.