HMMT 2025 U25

Let $A B C D$ be a trapezoid such that $A B \parallel C D$ , $A D = 13$ , $B C = 15$ , $A B = 20$ , and $C D = 34$ . Point $X$ lies inside the trapezoid such that $\angle X A B = 2\angle X B A$ and $\angle X D C = 2\angle X C D$ . Compute $X D - X A$ .