Question 10
Triangle $ABC$ is inscribed in circle $\omega.$ Points $P$ and $Q$ are on side $\overline{AB}$ with $AP \lt AQ.$ Rays $CP$ and $CQ$ meet $\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP = 4,$ $PQ = 3,$ $QB = 6,$ $BT = 5,$ and $AS = 7,$ then $ST = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$