
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 \ge 3$  is said to be nice if and only if there is at least one permutation  $ a_{1}, \cdots, a_{n}$  of  $ 1, \cdots, n$  such that the  $ n$  sums  $ a_{1} \minus{} a_{2} \plus{} a_{3} \minus{} \cdots \minus{} a_{n \minus{} 1} \plus{} a_{n}$ ,  $ a_{2} \minus{} a_{3} \plus{} a_{3} \minus{} \cdots \minus{} a_{n} \plus{} a_{1}$ ,  $ a_{3} \minus{} a_{4} \plus{} a_{5} \minus{} \cdots \minus{} a_{1} \plus{} a_{2}$ ,  $ \cdots$ ,  $ a_{n} \minus{} a_{1} \plus{} a_{2} \minus{} \cdots \minus{} a_{n \minus{} 2} \plus{} a_{n \minus{} 1}$  are all positive. Determine the set of all `nice' integers.

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