
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 the locus of the vertices of the parabolas given by the equation

$$
y = x^{2} - \frac{4 m}{1 + m^{2}} x + \frac{1 + 4 m^{2} - m^{4}}{\left(1 + m^{2}\right)^{2}}
$$

where \( m \) takes all real values.

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