Geometry: Power of a Point & Mass Points · Medium problem

In the coordinate plane the circle $\omega$ passes through $A=(0,0)$, $B=(6,0)$, and $C=(6,8)$. The point $P=(15,9)$ lies outside $\omega$, and a line through $P$ meets $\omega$ at two points $M$ and $N$ with $M$ between $P$ and $N$. Given that $PM=9$, find the length $MN$.