SHARYGIN 2019 Problem 6

Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD, AB_1CD$ and $ABC_1D$ are cyclic quadrilaterals. Prove that the quadrilateral $ABCD_1$ is also cyclic.