
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
Let \( A_{1} A_{2} \ldots A_{k} \) be a regular \( k \)-gon inscribed in a circle of radius 1, and let \( P \) be a point lying on or inside the circumcircle. Find the maximum possible value of \( \left(P A_{1}\right)\left(P A_{2}\right) \cdots\left(P A_{k}\right) \).

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