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

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