US-TST 2021 Problem 2

Points $A, V_{1}, V_{2}, B, U_{2}, U_{1}$ lie fixed on a circle $\Gamma$, in that order, and such that $B U_{2}>A U_{1}>B V_{2}>A V_{1}$. Let $X$ be a variable point on the $\operatorname{arc} V_{1} V_{2}$ of $\Gamma$ not containing $A$ or $B$. Line $X A$ meets line $U_{1} V_{1}$ at $C$, while line $X B$ meets line $U_{2} V_{2}$ at $D$. Prove there exists a fixed point $K$, independent of $X$, such that the power of $K$ to the circumcircle of $\triangle X C D$ is constant.