
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 a natural number  $k>1$ , define  $S_k$  to be the set of all triplets  $(n,a,b)$  of natural numbers, with  $n$  odd and  $\gcd (a,b)=1$ , such that  $a+b=k$  and  $n$  divides  $a^n+b^n$ . Find all values of  $k$  for which  $S_k$  is finite.

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