Judith is playing with triangles and circles

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Judith inscribed a circle in a equilateral triangle of sides \(x\) \(cm\). Then she inscribed another triangle in that circle. If Judith could do this process infinitely, the sum of the areas of all the triangles and circles would be expressed as \(\dfrac{x^{2}}{k}(\dfrac{\pi}{k}+\sqrt{k})\). Find \(k\).

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