HMMT 2014 U25

Let $A B C$ be an equilateral triangle of side length 6 inscribed in a circle $\omega$. Let $A_{1}, A_{2}$ be the points (distinct from $A$ ) where the lines through $A$ passing through the two trisection points of $B C$ meet $\omega$. Define $B_{1}, B_{2}, C_{1}, C_{2}$ similarly. Given that $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ appear on $\omega$ in that order, find the area of hexagon $A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$.