
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  $\times$  represent the cross product in  $\mathbb{R}^3.$  For what positive integers  $n$  does there exist a set  $S \subset \mathbb{R}^3$  with exactly  $n$  elements such that   $$ S=\{v \times w: v, w \in S\}? $$  

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