
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
An odd integer \( n \geq 3 \) is said to be "nice" if there is at least one permutation \( a_{1}, a_{2}, \ldots, a_{n} \) of \( 1, 2, \ldots, n \) such that the \( n \) sums 

\[ a_{1} - a_{2} + a_{3} - \cdots - a_{n-1} + a_{n}, \]
\[ a_{2} - a_{3} + a_{4} - \cdots - a_{n} + a_{1}, \]
\[ a_{3} - a_{4} + a_{5} - \cdots - a_{1} + a_{2}, \]
\[ \cdots, \]
\[ a_{n} - a_{1} + a_{2} - \cdots - a_{n-2} + a_{n-1} \]

are all positive. Determine the set of all "nice" integers.

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