TxO Math Bowl 2024 - Guts Contest - Set 7 Problem 1

An equilateral triangle $ABC$ with side length $\sqrt3$ is drawn in the plane. Two congruent circles, $\omega_1$ and $\omega_2$ are drawn internally tangent to the circumcircle of $ABC$ at $B$ and $C$, respectively. As well, $\omega_1$ and $\omega_2$ are externally tangent at some point. Circle $\omega_3$ is drawn in the interior of $ABC$ such that it is tangent to $AB,AC$ and externally tangent to $\omega_1,$ and $\omega_2$. The radius of $\omega_3$ can be written as $a-\sqrt{\sqrt b- c}$, where $a$, $b$, and $c$ are positive integers. Find the value of $a+b+c$.