A golden ring is to be cast in a circular mold. The circle circumscribes a square whose side \(A\), is the sum of two portions \(B\) and \(C\). The depth of the mold will be equal to \(C\).

Where \(B = 1 cm\), and \(\frac{A}{B} = \frac{B}{C}\), what will be the volume of the opening of the ring in cubic cm, assuming the inside is cylindrical? Round your answer to the nearest ten-thousandth.

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