Let $\triangle ABC$ be a triangle with $AB = 4$, $BC = 6$, and $CA = 5$. Let the angle bisector of $\angle BAC$ intersect $BC$ at the point $D$ and the circumcircle of $\triangle ABC$ again at the point $M \neq A$. The perpendicular bisector of segment $DM$ intersects the circle centered at $M$ passing through $B$ at two points, $X$ and $Y$. Compute $AX \cdot AY$.

Let's think step by step and output the final answer within \boxed{}.
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