
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 \( n \) satisfying the following property: if \( a_1, \ldots, a_n \) are integers whose sum is not divisible by \( n \), then there exists a permutation \( \left(b_i\right)_{1 \leq i \leq n} \) of \( \left(a_i\right)_{1 \leq i \leq n} \) such that \( n \) divides \( b_1 + 2b_2 + \cdots + n b_n \).

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