
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
The circle C passes through points M(5, 2) and N(3, 2), and its center is on the x-axis. Point A is a point on circle C, and O is the origin of the coordinates.
(1) Find the equation of circle C;
(2) Connect OA, extend OA to P so that |OA| = |AP|, and find the equation of the trajectory of point P.

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