Nemotron-IMO-Bench 2026 G50

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ with circumcenter $O_1$ meet at point $P$. If the four incenters of triangles $PAB, PBC, PCD, PDA$ lie on a circle with centre $O_2$, prove that $P, O_1, O_2$ are collinear.