\(\Gamma\) is a circle with a diameter \(BC\), \(A\) is a point on \(BC\) such that \(AB = 3\) and \(AC = 9,\) and \(D\) is a point on circle \(\Gamma\) such that \(\overline{AD} \perp \overline{BC}.\) The tangents to circle \(\Gamma\) at points \(B\) and \(D\) intersect at \(E.\) The line from \(E\) that is parallel to line \(BC\) intersects the circle at points \(F\) and \(G,\) such that \(F\) is between \(E\) and \(G.\) The area of \(\triangle CGE\) can be expressed in the form \(\sqrt{a} + \sqrt{b},\) where \(a \) and \(b\) are positive integers. What is the value of \(a+b?\)

This problem is posed by Michael T.

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