
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 \( d \) be a non-perfect square positive integer. Suppose \( x_0, y_0, x, \) and \( y \) are all positive integers, and \( X \) and \( Y \) are integers. If \( X + Y \cdot \sqrt{d} = (x + y \cdot \sqrt{d}) \cdot (x_0 - y_0 \cdot \sqrt{d})^r \), then find \( X - Y \cdot \sqrt{d} \).

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