SHARYGIN 2006 Problem 11

In the triangle $ABC, O$ is the center of the circumscribed circle, $A ', B', C '$ are the symmetrics of $A, B, C$ with respect to opposite sides, $A_1, B_1, C_1$ are the intersection points of the lines $OA'$ and $BC, OB'$ and $AC, OC'$ and $AB$ . Prove that the lines $A A_1, BB_1, CC_1$ intersect at one point.