
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
Definition: If for any point $(x\_0, y\_0)$ in the plane point set $A$, there always exists a positive real number $r$ such that the set ${(x, y) | \sqrt{(x - x\_0)^2 + (y - y\_0)^2} < r} \subseteq A$, then $A$ is called an open set. Determine which of the following sets are not open sets:

1. ${(x, y) | x^2 + y^2 = 1}$;
2. ${(x, y) | x + y + 2 > 0}$;
3. ${(x, y) | |x + y| \leq 6}$;
4. ${(x, y) | 0 < x^2 + (y - \sqrt{2})^2 < 1}$.

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