HMMT 2025 U29

Points $A$ and $B$ lie on circle $\omega$ with center $O$ . Let $X$ be a point inside $\omega$ . Suppose that $XO = 2\sqrt{2}$ , $XA = 1$ , $XB = 3$ , and $\angle AXB = 90^{\circ}$ . Points $Y$ and $Z$ are on $\omega$ such that $Y \neq A$ and triangles $\triangle AXB$ and $\triangle YXZ$ are similar with the same orientation. Compute $XY$ .