Judith is playing with triangles and circles

Level pending

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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