RU-239-OPEN 2021 Problem 2

A triangle $ABC$ with an obtuse angle at the vertex $C$ is inscribed in a circle with a center at point $O$ . Circumcircle of triangle $AOB$ centered at point $P$ intersects line $AC$ at points $A$ and $A_1$ , line $BC$ at points $B$ and $B_1$ , and the perpendicular bisector of the segment $PC$ at points $D$ and $E$ . Prove that points $D$ and $E$ together with the centers of the circumscribed circles of triangles $A_1OC$ and $B_1OC$ lie on one circle.