JBMO-SL 2004 G3

Let $A B C$ be scalene triangle inscribed in the circle $k$. Circles $\alpha, \beta, \gamma$ are internally tangent to $k$ at points $A_{1}, B_{1}, C_{1}$ respectively, and tangent to the sides $B C, C A, A B$ at points $A_{2}, B_{2}, C_{2}$ respectively, so that $A$ and $A_{1}$ are on opposite sides of $B C^{\prime}, B$ and $B_{1}$ mre on opposite sides of $C A$, and $C$ and $C_{1}$ are on opposite sides of $A B$. Lines $A_{1} A_{2}, B_{1} B_{2}$ and $C_{1} C_{2}$ meet again the circle $k$ in the points $A^{\prime}, B^{\prime}, C^{\prime \prime}$ respectively. is right.

Prove that if $M$ is the intersection point of the lines $B B^{\prime}$ and $C^{\prime} C^{\prime \prime}$, the angle $\angle M A .4^{\prime}$