HMMT November 2024 U14

Let $A B C D$ be a trapezoid with $A B \| C D$. Point $X$ is placed on segment $\overline{B C}$ such that $\angle B A X=$ $\angle X D C$. Given that $A B=5, B X=3, C X=4$, and $C D=12$, compute $A X$.