
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) A and B move clockwise with equal angular speed along circles centered at P and Q, respectively. C moves continuously so that AB = BC = CA. Establish C's locus and speed.

(b) ABC is an equilateral triangle and P satisfies AP = 2, BP = 3. Establish the maximum possible value of CP.

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