
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
Let  $\mathbb{Z}^n$  be the integer lattice in  $\mathbb{R}^n$ . Two points in  $\mathbb{Z}^n$  are called {\em neighbors} if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers  $n \geq 1$  does there exist a set of points  $S \subset \mathbb{Z}^n$  satisfying the following two conditions? 
(1) If  $p$  is in  $S$ , then none of the neighbors of  $p$  is in  $S$ . 
(2) If  $p \in \mathbb{Z}^n$  is not in  $S$ , then exactly one of the neighbors of  $p$  is in  $S$ .


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