Question 86
Let triangle $ABC$ have $AB = 5$, $BC = 8$, and $\angle ABC = 60^{\circ}$. A circle $\omega$ tangent to segments $\overline{AB}$ and $\overline{BC}$ intersects segment $\overline{CA}$ at points $X$ and $Y$ such that points $C, Y, X$, and $A$ lie along $\overline{CA}$ in this order. If $\omega$ is tangent to $\overline{AB}$ at point $Z$ and $ZY \parallel BC$, compute the radius of $\omega$.