
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  $\triangle P_1P_2P_3$  be an equilateral triangle. For each  $n\ge 4$ , *Mingmingsan* can set  $P_n$  as the circumcenter or orthocenter of  $\triangle P_{n-3}P_{n-2}P_{n-1}$ . Find all positive integer  $n$  such that *Mingmingsan* has a strategy to make  $P_n$  equals to the circumcenter of  $\triangle P_1P_2P_3$ .  

*Proposed by Li4 and Untro368.*

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