Selection Examination

Let $AB\Gamma$ be a triangle with circumcircle $\omega (O,R)$. A circle $\gamma$ passes through $O$ and $B$ and is tangent to the line $AB$ at $B$. Let the circle $\kappa$ meets the circle $\omega (O,R)$ for a second time at $P\ne B$. A circle passes through $P$ and $\Gamma$ and is tangent to the line $A\Gamma$ at $\Gamma$ and meets the circle $\gamma$ at $M\ne P$. Prove that: $MP=M\Gamma$.