
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
Find all positive integers $n$ $(n \geqslant 3)$ for which there exists a set $M$ with $n$ elements, where the elements are distinct nonzero vectors of the same length, such that $\sum_{u \in M} u = 0$ and for any $v, w \in M$, we have $\boldsymbol{v}+\boldsymbol{w} \neq \mathbf{0}$.

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