Regular Polygons are arranged around a unit equilateral triangle as shown in the figure above.

If the hatched area can be written as \[\frac{1}{d} \frac{\sqrt{a}-\sqrt{b}}{\sqrt{3d(\sqrt{c}+c)}+\sqrt{c}-1}\] for square-free integers \(a, b, c\) and \(d\). Find \(a+b+c+d\).

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