HMMT 2009 F8

The incircle $\omega$ of equilateral triangle $A B C$ has radius 1. Three smaller circles are inscribed tangent to $\omega$ and the sides of $A B C$, as shown. Three smaller circles are then inscribed tangent to the previous circles and to each of two sides of $A B C$. This process is repeated an infinite number of times. What is the total length of the circumferences of all the circles?