HMMT 2025 U23

Regular hexagon $ABCDEF$ has side length 2. Circle $\omega$ lies inside the hexagon and is tangent to segments $\overline{AB}$ and $\overline{AF}$ . There exist two perpendicular lines tangent to $\omega$ that pass through $C$ and $E$ , respectively. Given that these two lines do not intersect on line $AD$ , compute the radius of $\omega$ .