
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
It is obvious that \( x = y = 0 \) are the roots of this equation. Let \( x, y > 0 \) be natural numbers greater than or equal to 1. Write the equation in the form: \((\sqrt{2x} - 1)(\sqrt{2y} - 1) = 1\). Assume for definiteness that \( y \geq x \), then \(\sqrt{2x} \leq 1\), i.e., \( 2x \leq 4 \), \( x \leq 2 \Rightarrow x = 1 \) or \( x = 2 \). For \( x = 1 \), we obtain: \((\sqrt{2} - 1)(\sqrt{2y} - 1) = 1 \Rightarrow y = 3 + \sqrt{2}\) - a non-integer. For \( x = 2 \), we obtain \( y = 2 \), hence \( x = y = 2 \) is the only solution that is strictly greater than zero.

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