A circle having common centre with the circumcircle of triangle $ABC$ meets the sides of the triangle at six points forming convex hexagon $A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$ ($A_{1}$ and $A_{2}$ lie on $BC$, $B_{1}$ and $B_{2}$ lie on $AC$, $C_{1}$ and $C_{2}$ lie on $AB$). If $A_{1}B_{1}$ is parallel to the bisector of angle $B$, prove that $A_{2}C_{2}$ is parallel to the bisector of angle $C$. Proposed by S. Berlov