
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
Find all positive integer  $n$ , such that there exists  $n$  points  $P_1,\ldots,P_n$  on the unit circle , satisfying the condition that for any point  $M$  on the unit circle,  $\sum_{i=1}^n MP_i^k$  is a fixed value for 
a)  $k=2018$ b)  $k=2019$ .

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