
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 positive integers  $n$  such that there exists a prime number  $p$ , such that  $p^n-(p-1)^n$  is a power of  $3$ .

Note. A power of  $3$  is a number of the form  $3^a$  where  $a$  is a positive integer.

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