The points $A, B, C, D$ lie on the line $\ell$ in this order. The points $P$ and $Q$ are chosen on one side of the line $\ell$, and the point $R$ is chosen on the other side so that: $$\angle APB = \angle CPD = \angle QBC = \angle QCB = \angle RAD = \angle RDA$$ Prove that the points $P, Q, R$ lie on the same line. Proposed by Mykhailo Shtandenko, Fedir Yudin