
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
Determine \( y \) such that the roots \( x_1 \) and \( x_2 \) of the equation

\[ x^{2} - 3 x y + y^{2} + 2 x - 9 y + 1 = 0 \]

are equal; for which values of \( y \) are the roots of the equation real? Also, solve the equation for \( y \) and determine \( x \) such that the roots of the equation are real.

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