
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 number \( n \geqslant 3 \) is called "good" if and only if there exists a permutation \(\{a_{1}, a_{2}, \cdots, a_{n}\}\) of \(\{1,2, \cdots, n\}\) such that the following \( n \) sums are all positive:
\[ a_{1}-a_{2}+a_{3}-a_{4}+\cdots-a_{n-1}+a_{n}, \]
\[ a_{2}-a_{3}+a_{4}-a_{5}+\cdots-a_{n}+a_{1}, \]
\[ \cdots \cdots \]
\[ a_{n}-a_{1}+a_{2}-a_{3}+\cdots-a_{n-2}+a_{n-1}. \]

Determine all "good" odd numbers.

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