
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 an integer \( n > 1 \), let \( a_{1}, a_{2}, \cdots, a_{n} \) be distinct non-negative real numbers. Define the sets
\[ 
A = \left\{ a_{i} + a_{j} \mid 1 \leq i \leq j \leq n \right\}, \quad B = \left\{ a_{i} a_{j} \mid 1 \leq i \leq j \leq n \right\}.
\]
Find the minimum value of \( \frac{|A|}{|B|} \), where \( |X| \) denotes the number of elements in the set \( X \).

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