
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 triples  $(p, x, y)$  consisting of a prime number  $p$  and two positive integers  $x$  and  $y$  such that  $x^{p -1} + y$  and  $x + y^ {p -1}$  are both powers of  $p$ .

*Proposed by Belgium*

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