
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
Which prime numbers \( p \) satisfy the system of equations

\[
p + 1 = 2x^2 \\
p^2 + 1 = 2y^2
\]

where \( x \) and \( y \) are integers?

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