Suppose $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB = 10$ and $CD = 15$, compute the maximum possible value of $XY$.

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