
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
Let  $S_r(n)=1^r+2^r+\cdots+n^r$  where  $n$  is a positive integer and  $r$  is a rational number. If  $S_a(n)=(S_b(n))^c$  for all positive integers  $n$  where  $a, b$  are positive rationals and  $c$  is positive integer then we call  $(a,b,c)$  as *nice triple.* Find all nice triples.

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