
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
The *cross* of a convex  $n$ -gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a  $3 \times 4$  rectangle is  $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$ .

Suppose  $S$  is a dodecagon ( $12$ -gon) inscribed in a unit circle. Find the greatest possible cross of  $S$ .

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