Consider a fixed circle $\Gamma$ with three fixed points $A$, $B$, and $C$ on it. Also let us fix a real number $\lambda \in (0,1)$. For a variable point $P \notin \{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM = \lambda \cdot CP$. Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$.

Prove that as $P$ varies, the point $Q$ lies on a fixed circle.

(IMO-2014 Shortlist, Problem G4)