
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
For every positive integer \(n\), determine the number of permutations \(\left(a_{1}, a_{2}, \ldots, a_{n}\right)\) of the set \(\{1,2, \ldots, n\}\) with the following property:
\[ 2\left(a_{1}+a_{2}+\cdots+a_{k}\right) \text{ is divisible by } k \text{ for } k = 1, 2, \ldots, n. \]

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