SHARYGIN 2010 Problem 3

Points $A', B', C'$ lie on sides $BC, CA, AB$ of a triangle $ABC$. For a point $X$ one has $\angle AXB = \angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' + \angle BAC$. Prove that the quadrilateral $XA'BC'$ is cyclic.