A circular disk of radius \(R\) consists of two uniform halves of equal area with masses \(\frac{2}{3}M\) and \(\frac{1}{3}M\). The disk is free to rotate on an axle through its geometric center.

Initially, the disk is at rest as shown on the left such that the half of mass \(\frac23 M\) is on top.

Suppose the disk is toppled by an extremely gentle nudge. At the instant the disk first reaches the orientation on the right, its angular speed can be expressed as follows:

\[\omega = \sqrt{\frac{Ag}{B \pi R}}. \]

If \(A\) and \(B\) are coprime positive integers, determine \(A + B\).

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