
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
For a set \( Y \) consisting of a finite number of real numbers, define \( \sigma(Y) \) as the sum of all elements in \( Y \): \(\sigma(Y) = \sum_{y \in Y} y \). Given positive integers \( m \), \( n \) and positive real numbers \( x_{1}<x_{2}<\cdots<x_{m} \), let \( A_{1}, A_{2}, \cdots, A_{n} \) be non-empty subsets of the set \(\{x_{1}, x_{2}, \cdots, x_{m}\}\). Find the minimum value of the expression
$$
\sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\sigma\left(A_{i} \cap A_{j}\right)}{\sigma\left(A_{i}\right) \sigma\left(A_{j}\right)}
$$

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