TUYMAADA 2020 S6

An isosceles triangle $ABC$ ( $AB = BC$ ) is given. Circles $\omega_1$ and $\omega_2$ with centres $O_1$ and $O_2$ lie in the angle $ABC$ and touch the sides $AB$ and $CB$ at $A$ and $C$ respectively, and touch each other externally at point $X$ . The side $AC$ meets the circles again at points $Y$ and $Z$ . $O$ is the circumcenter of the triangle $XYZ$ . Lines $O_2 O$ and $O_1 O$ intersect lines $AB$ and $BC$ at points $C_1$ and $A_1$ respectively. Prove that $B$ is the circumcentre of the triangle $A_1 OC_1$ .