
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 \( x_{0} + \sqrt{2003} y_{0} \) be the fundamental solution of the equation \( x^{2} - 2003 y^{2} = 1 \). Find the solution \((x, y)\) of the equation such that \( x, y > 0 \) and all prime factors of \( x \) divide \( x_{0} \).

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