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

An equilateral triangle ABC with side length \sqrt{3} 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.