Advanced Number Theory · Challenge problem

Equilateral triangle $ABC$ has side $6$. Its inscribed circle is drawn. Then, in each of the three corners, a circle is drawn tangent to the two sides at that corner and to the inscribed circle; then, in each corner, another circle tangent to the two sides and to the previous corner circle; and so on forever. The total area of all the circles drawn (the inscribed circle and the three infinite chains) is $\dfrac{m}{n}\pi$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.