HMMT November 2022 T6

A triangle $X Y Z$ and a circle $\omega$ of radius 2 are given in a plane, such that $\omega$ intersects segment $\overline{X Y}$ at the points $A, B$, segment $\overline{Y Z}$ at the points $C, D$, and segment $\overline{Z X}$ at the points $E, F$. Suppose that $X B>X A, Y D>Y C$, and $Z F>Z E$. In addition, $X A=1, Y C=2, Z E=3$, and $A B=C D=E F$. Compute $A B$.