
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
Let \( n \) be a natural number. Determine the number of permutations \(\left(a_{1}, \ldots, a_{n}\right)\) of the set \(\{1,2, \ldots, n\}\) with the following property:

\[ 
2\left(a_{1}+\ldots+a_{k}\right) \quad \text{is divisible by } k \quad \forall k \in\{1,2, \ldots, n\} 
\]

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