A Strange Solid

Level pending

A solid has a circular base with radius \(1.\) Parallel cross-sections perpendicular to the base are equilateral triangles.

The volume of the solid can be represented by \(\dfrac{A\sqrt{B}}{C}\text{,}\) where \(A\) and \(C\) are positive coprime integers and \(B\) is square-free. What is \(A+B+C\text{ ?}\)

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