
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
Given a point F(-2, 0) and a point G on the circle $$C_{1}: (x+4)^{2}+y^{2}=16.$$  
(1) If the line FG intersects the line x=-4 at point T, and G is the midpoint of segment GT, find the length of the chord cut by line FG on circle C.  
(2) Does there exist a fixed point P in the plane such that |GP|=2|GF|? If it exists, find the coordinates of P; if not, provide the reason.

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