
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
Given any \( h = 2^r \) (where \( r \) is a non-negative integer), find all natural numbers \( k \) that satisfy the following conditions: For each such \( k \), there exist an odd natural number \( m > 1 \) and a natural number \( n \) such that \( k \) divides \( m^k - 1 \) and \( m \) divides \( n^{\frac{n^k - 1}{k}} + 1 \).

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