Of circles, tangents, and triangles?

Geometry Level 5

In the figure above, circle \(B\) is tangential to circle \(A\) at \(G\). \( \overline {AG} \) is perpendicular to \( \overline {CD} \), intersecting at \(B\). The length of \( \overline {CD} \) is three times the diameter of circle \(B\). Radii \(AF\) and \(AE\) are drawn such that they are tangential to circle \(B\). If \( \angle FAE \) can be expressed in the form of \( \cos^{-1} ( \frac{a}{b} ) \) where \(a\) and \(b\) are coprime positive integers,

determine \(a \times b\).

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