HMMT 2025 T3

Let $\omega_{1}$ and $\omega_{2}$ be two circles intersecting at distinct points $A$ and $B$ . Point $X$ varies along $\omega_{1}$ , and point $Y$ on $\omega_{2}$ is chosen such that $AB$ bisects the angle $\angle XAY$ . Prove that as $X$ varies along $\omega_{1}$ , the circumcenter of $\triangle AXY$ (if it exists) varies along a fixed line.