
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
For which real values of \( x \) and \( y \) do the complex numbers \( z_{1} = 2x^{2} - 3i - 1 + yi \) and \( z_{2} = y + x^{2} i - 3 - 2i \) become conjugates of each other?

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