
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
Find the geometric locus of points in the coordinate plane whose coordinates \(x\) and \(y\) satisfy one of the following conditions:
1. \(y^{2} - x^{2} = 0\)
2. \(x^{2} + y^{2} = 4(y - 1)\)
3. \(x^{2} - 2x + 1 = 0\)
4. \(x^{2} - 2xy + y^{2} = -1\).

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