HMMT November 2024 U32

Let $A B C$ be an acute triangle and $D$ be the foot of altitude from $A$ to $\overline{B C}$. Let $X$ and $Y$ be points on the segment $\overline{B C}$ such that $\angle B A X=\angle Y A C, B X=2, X Y=6$, and $Y C=3$. Given that $A D=12$, compute $B D$.