SHARYGIN 2010 Problem 4

The diagonals of a cyclic quadrilateral $ABCD$ meet in a point $N$. The circumcircles of the triangles $ANB$ and $CND$ intersect the sidelines $BC$ and $AD$ for the second time in points $A_1, B_1, C_1, D_1$. Prove that the quadrilateral $A_1B_1C_1D_1$ is inscribed into a circle centered at $N$.