In a circle \(A\) with radius \(\frac{18+\sqrt{432}}{6}\) we draw \(3\) circles with equal radii, in such way that each of them are tangent to the other \(2\) circles and to circle \(A\). Inside the circle \(A\) we draw a circle \(B\) in such way that is tangent to each of the \(3\) new circles. An equilateral triangle \(C\) is inscribed in \(B\).

If the area of \(C\) can be written in the form:

\(\dfrac{a\sqrt{b}-c}{d}\)

Where \(a,b,c\) and \(d\) are integers such that \(b|a\) and \(d|c\).

Find:

\(a+b+c+d\)

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