
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the Cartesian coordinate system $xOy$, points $E$ and $F$ have coordinates $(0, 1)$ and $(0, -1)$, respectively. A moving point $G$ satisfies the condition that the product of the slopes of lines $EG$ and $FG$ is $- \frac {1}{4}$.  
(1) Find the equation of the trajectory of point $G$;  
(2) Suppose $A$ and $B$ are the left and right vertices of the trajectory of point $G$, and $P$ is a moving point on the line $l: x=4$ (point $P$ is not on the x-axis). Connect $AP$ and intersect the trajectory of $G$ at point $C$, connect $PB$ and extend it to intersect the trajectory of $G$ at point $D$. Is line $CD$ passing through a fixed point? If so, find the coordinates of this fixed point; if not, explain why.

Present the answer in LaTex format: \boxed{Your answer}
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