SHARYGIN 2020 Problem 17

Chords $A_1A_2$ and $B_1B_2$ meet at point $D$ . Suppose $D'$ is the inversion image of $D$ and the line $A_1B_1$ meets the perpendicular bisector to $DD'$ at a point $C$ . Prove that $CD\parallel A_2B_2$ .