
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
Do there exist positive integers  $a>b>1$  such that for each positive integer  $k$  there exists a positive integer  $n$  for which  $an+b$  is a  $k$ -th power of a positive integer?

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