
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
A finite set of lattice points \( S \) in the plane is called a double-neighbor set if for every point \((p, q)\) in \( S \), exactly two of the points \((p+1, q)\), \((p, q+1)\), \((p-1, q)\), and \((p, q-1)\) are in \( S \). For which positive integers \( n \) does there exist a double-neighbor set containing exactly \( n \) lattice points?

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