
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 the positive integer \( k > 1 \), define \( S_{k} \) as the set of all tuples of positive integers \((n, a, b)\) that satisfy the following conditions: \( n \) is an odd number, \( \gcd(a, b)=1 \), \( a + b = k \), and \( n \mid \left(a^{n} + b^{n}\right) \). Determine all values of \( k \) such that \( S_{k} \) is a finite set.

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