
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 \( V \) be the set of vertices of a regular 21-gon. Given a non-empty subset \( U \) of \( V \), let \( m(U) \) be the number of distinct lengths that occur between two distinct vertices in \( U \). What is the maximum value of \( \frac{m(U)}{|U|} \) as \( U \) varies over all non-empty subsets of \( V \)?

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