
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
Given a non-empty subset family \( U \) of \( S = \{a_1, a_2, \ldots, a_n\} \) that satisfies the property: if \( A \in U \) and \( A \subseteq B \), then \( B \in U \); and a non-empty subset family \( V \) of \( S \) that satisfies the property: if \( A \in V \) and \( A \supseteq B \), then \( B \in V \). Find the maximum possible value of \( \frac{|U \cap V|}{|U| \cdot |V|} \).

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