Advanced Number Theory · Superchallenge problem

Triangle $ABC$ is isosceles with $AB = AC$. Circles $\omega_1, \omega_2, \omega_3, \ldots$ all lie inside the triangle: $\omega_1$ is the incircle, and for each $n \ge 1$, $\omega_{n+1}$ is tangent to sides $AB$ and $AC$ and externally tangent to $\omega_n$, lying between $\omega_n$ and $A$. The total circumference of all the circles is $\sqrt{3}\pi$, and their total area is $\dfrac{3\pi}{8}$. The area of the region inside triangle $ABC$ but outside every circle can be written as $\sqrt{a} - \dfrac{b\pi}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. Find $a + b + c$.