
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  $ABC$  be an equilateral triangle, and let  $P$  be some point in its circumcircle.
Determine all positive integers  $n$ , for which the value of the sum $S_n (P) = |PA|^n +  |PB|^n +  |PC|^n$  is independent of the choice of point  $P$ .

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