HMMT November 2024 E7

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