Trigonometry · Superchallenge problem

A circular sector has radius $12$ and central angle $60^\circ$. Circle $\omega_1$ is tangent to both straight sides of the sector and to its arc. Circle $\omega_2$ is tangent to both straight sides and to $\omega_1$, lying between $\omega_1$ and the vertex; circle $\omega_3$ is tangent to both sides and to $\omega_2$, and so on forever. Find the total area of all the circles.