Given a sector of radius 10 and angle \( \frac{ \pi}{3} \), we draw the largest circle (in green) that is tangent to all sides of the sector. We then draw the circle (in pink) that is tangent to the 2 sides, and also to the green circle. We repeat this process indefinately, where the next circle is shown in blue.

Find the sum of the areas of these infinitely many circles.

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