BALTIC-WAY 2023 Problem 15

Let $\omega_{1}$ and $\omega_{2}$ be circles with no common points, such that neither circle lies inside the other. Points $M$ and $N$ are chosen on the circles $\omega_{1}$ and $\omega_{2}$, respectively, such that the tangent to the circle $\omega_{1}$ at $M$ and the tangent to the circle $\omega_{2}$ at $N$ intersect at $P$ and such that $P M N$ is an isosceles triangle with $P M=P N$. The circles $\omega_{1}$ and $\omega_{2}$ meet the segment $M N$ again at $A$ and $B$, respectively. The line $P A$ meets the circle $\omega_{1}$ again at $C$ and the line $P B$ meets the circle $\omega_{2}$ again at $D$. Prove that $\angle B C N=\angle A D M$.