
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 which integers \( n > 1 \) have the property that there exists an infinite sequence \( a_{1}, a_{2}, a_{3}, \ldots \) of nonzero integers such that the equality
\[ a_{k} + 2 a_{2k} + \cdots + n a_{nk} = 0 \]
holds for every positive integer \( k \).

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