
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
Consider all circles on the plane whose equations can be written in the form

$$
\alpha\left((x-2)^{2}+y^{2}-1\right)+\beta\left((x+2)^{2}+y^{2}-1\right)=0
$$

where $\alpha$ and $\beta$ are real numbers. Where are the points that do not lie on any of these circles?

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