Let $A , B$ be the left and right vertices of the ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + y ^ { 2 } = 1 ( a > 1 )$ respectively, $G$ be the upper vertex of $E$ , and $\overrightarrow { A G } \cdot \overrightarrow { G B } = 8$ . $P$ is a moving point on the line $x = 6$ , the other intersection point of $P A$ with $E$ is $C$ , and the other intersection point of $P B$ with $E$ is $D$ .
(1) Find the equation of $E$ ;
(2) Prove that the line $C D$ passes through a fixed point.

Equation of the line passing through the points of intersection of the parabola $x ^ { 2 } = 8 y$ and the ellipse $\frac { x ^ { 2 } } { 3 } + y ^ { 2 } = 1$ is :
(1) $y - 3 = 0$
(2) $y + 3 = 0$
(3) $3 y + 1 = 0$
(4) $3 y - 1 = 0$