Michael starts with ABC

$\Gamma$ 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 .