
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 \( n \) be a positive integer and let \( p \) be a prime number, \( p > 3 \). Find at least \( 3(n+1) \) (easier version: \( 2(n+1) \)) sequences of positive integers \( x, y, z \) satisfying
\[ 
xyz = p^n (x + y + z) 
\]
that do not differ only by permutation.

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