Equilateral triangle \triangle A B C is inscribed in circle \omega with radius 18. Circle \omega_{A} is tangent to sides \overline{A B} and \overline{A C} and is internally tangent to \omega. Circles \omega_{B} and \omega_{C} are defined analogously. Circles \omega_{A}, \omega_{B}, and \omega_{C} meet in six points-two points for each pair of circles. The three intersection points closest to the vertices of \triangle A B C are the vertices of a large equilateral triangle in the interior of \triangle A B C, and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of \triangle A B C. The side length of the smaller equilateral triangle can be written as \sqrt{a}-\sqrt{b}, where a and b are positive integers. Find a+b.
