
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
Determine all positive integers  $n\geq 2$  for which there exist integers  $x_1,x_2,\ldots ,x_{n-1}$  satisfying the condition that if  $0<i<n,0<j<n, i\neq j$  and  $n$  divides  $2i+j$ , then  $x_i<x_j$ .

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