
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