
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 $F(-1,0)$ and line $l:x=-4$, if the distance $d$ between moving point $P$ and line $l$ satisfies $d=2|PF|$,

(1) Find the equation of the trajectory $C$ of point $P$;

(2) If line $PF$ intersects trajectory $C$ at another point $Q$, and $\overrightarrow{PF}=2\overrightarrow{FQ}$, a circle with center $P$ and radius $r=2|PQ|$ is intersected by line $l$ to form a chord $AB$. Find $|AB|$.

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