
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 integers \( n \geq 2 \) such that for any integers \( a_1, a_2, \ldots, a_n \) which are not all divisible by \( n \), there always exists an index \( i \in\{1, 2, \ldots, n\} \) such that none of the sums \( a_i, a_i + a_{i+1}, \ldots, a_i + a_{i+1} + \ldots + a_{i+n-1} \) (with indices considered modulo \( n \)) are divisible by \( n \).

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