SHARYGIN 2018 Problem 18

Let $C_1, A_1, B_1$ be points on sides $AB, BC, CA$ of triangle $ABC$ , such that $AA_1, BB_1, CC_1$ concur. The rays $B_1A_1$ and $B_1C_1$ meet the circumcircle of the triangle at points $A_2$ and $C_2$ respectively. Prove that $A, C$ , the common point of $A_2C_2$ and $BB_1$ and the midpoint of $A_2C_2$ are concyclic.