
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  $n$  be a positive integer and  $P_1, P_2, \ldots, P_n$  be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances  $P_iP_1, P_iP_2, \ldots, P_iP_n$  forms the same sequence  for all  $i=1,2, \ldots, n$  when these numbers are arranged in a non-decreasing order. Find all possible values of  $n$ .

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