
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 the parabola $C: y^2 = 4x$ with focus $F$.  
(1) Points $A$ and $P$ satisfy $\overrightarrow{AP} = -2\overrightarrow{FA}$. When point $A$ moves on parabola $C$, find the trajectory equation of the moving point $P$;  
(2) Is there a point $Q$ on the x-axis such that the symmetric point of $Q$ with respect to the line $y=2x$ is on parabola $C$? If it exists, find the coordinates of all such points $Q$; if not, please explain why.

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