
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
Find the smallest natural number  $ k$  for which there exists natural numbers  $ m$  and  $ n$  such that  $ 1324 \plus{} 279m \plus{} 5^n$  is  $ k$ -th power of some natural number.

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