
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  $n$  be a positive integer number. If  $S$  is a finite set of vectors in the plane, let  $N(S)$  denote the number of two-element subsets  $\{\mathbf{v}, \mathbf{v'}\}$  of  $S$  such that
\[4\,(\mathbf{v} \cdot \mathbf{v'}) + (|\mathbf{v}|^2 - 1)(|\mathbf{v'}|^2 - 1) < 0. \] 
Determine the maximum of  $N(S)$  when  $S$  runs through all  $n$ -element sets of vectors in the plane. 

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