UZ-MO 2014 Problem 5

Let $PA_1A_2...A_{12}$ be the regular pyramid, $A_1A_2...A_{12}$ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12}$ and $PA_1A_5$ is equal to $\alpha$ . Find the volume of the pyramid.