Circle, triangle, circle, triangle, etc

Geometry Level 4

A circle with a radius of \(3 \text{ m}\) has an equilateral triangle inscribed within it. Inside of this equilateral triangle lies another circle tangent to the triangle's edges. Within this circle lies another equilateral triangle. This pattern of inscription goes on to infinity, as shown in the figure.

If the area of the dark blue region can be expressed as \(A\pi -B\sqrt { C } \text{ m}^2\), where \(A\), \(B\), and \(C\) are positive integers and the radical is simplified as much as possible, find the value of the following expression: \[\frac { A+B+C }{ C }\]


This problem is partially inspired by Isaac Reid.

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