BD-BDMO 2021 Problem 6

Let $ABC$ be an acute-angled triangle. The external bisector of $\angle BAC$ meets the line $BC$ at point $N$ . Let $M$ be the midpoint of $BC$ . $P$ and $Q$ are two points on line $AN$ such that, $\angle PMN=\angle MQN=90^{\circ}$ . If $PN=5$ and $BC=3$ , then the length $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are co-prime positive integers. What is the value of $(a+b)$ ?