
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
Given the equation \( \sqrt{a(x-a)} + \sqrt{a(y-a)} = \sqrt{x-a} - \sqrt{a-y} \) holds true within the range of real numbers, where \( a, x, y \) are three distinct real numbers. Find the value of \( \frac{3x^2 + xy - y^2}{x^2 - xy + y^2} \).

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