Find the area of the hatched region, if the radii of the two circles in the figure above are \(1\) and \(\sqrt[5]{162+27\sqrt[5]{162+27\sqrt[5]{\cdots}}}\).

If your answer is in the form of \(a\sqrt{b}-\frac{c}{2b}\pi\) for some positive pairwise coprime integers \(a, b, c\). Find \(a+b+c\).

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