IMO 2021 Problem 4

Let $\Gamma$ be a circle with centre $I$ , and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$ . Let $\Omega$ be the circumcircle of the triangle $A I C$ . The extension of $B A$ beyond $A$ meets $\Omega$ at $X$ , and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$ . The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$ , respectively. Prove that $$A D+D T+T X+X A=C D+D Y+Y Z+Z C.$$

Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland