
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 all pairs of natural numbers \( n \) and \( k \) such that \( (n+1)^{k} = n! + 1 \). (As usual, \( n! \) denotes the product of all natural numbers up to \( n \). For example, \( 4! = 1 \cdot 2 \cdot 3 \cdot 4 \).)

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