
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
A moving circle P is externally tangent to the circle F: $(x-2)^2+y^2=1$ and is tangent to the line $x=-1$.
(1) Find the equation of the trajectory C of the center P of the moving circle.
(2) Are there two points A and B on the trajectory C that are symmetric about the line $y=x-1$? If so, find the coordinates of the two points. If not, explain the reason.

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