
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 integers $n$ such that for any sequence $\left(x_{i}\right)_{1 \leqslant i \leqslant n}$ where $n$ does not divide their sum, there exists $j \in\{1, \ldots, n\}$ such that $n$ does not divide $x_{j}, x_{j}+x_{j+1}, \ldots, x_{j}+\cdots+x_{j+n-1}$ (where the indices are taken modulo $n$).

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