KöMaL 2016 A685

Let $AB$ and $CD$ be two chords in a circle $\Omega$. Choose a circle $\omega$ tangent to the segments $AB$ and $CD$ at their interior points $M$ and $N$ respectively, and let $\omega$ intersect $\Omega$ at $P$ and $Q$. Suppose that a third circle $\omega'$, distinct from $\Omega$ and $\omega$, and also passing through $P$ and $Q$, meets the line $MN$ at $M'$ and $N'$. Prove that the points $A$, $B$, $C$, $D$, $M'$, and $N'$ lie on a conic section that is tangent to $\omega'$ at $M'$ and $N'$.