MathNet 2014 Question 40

Let $ABC$ 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 $BC$ 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 $A_1A_2B_1B_2C_1C_2$.



Supplemental illustration of the given configuration.