HMMT November 2020 E7

In triangle $A B C$ with $A B=8$ and $A C=10$, the incenter $I$ is reflected across side $A B$ to point $X$ and across side $A C$ to point $Y$. Given that segment $X Y$ bisects $A I$, compute $B C^{2}$. (The incenter $I$ is the center of the inscribed circle of triangle $A B C$.)