
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 \( d(n) \) denote the number of positive divisors of \( n \). Find all triples \( (n, k, p) \), where \( n \) and \( k \) are positive integers and \( p \) is a prime number, such that

\[ 
n^{d(n)} - 1 = p^k 
\]

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