
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 \( n > 2 \) be an integer and let \(\ell \in \{1, 2, \ldots, n\}\). A collection \( A_{1}, \ldots, A_{k} \) of (not necessarily distinct) subsets of \(\{1, 2, \ldots, n\}\) is called \(\ell\)-large if \( |A_{i}| \geq \ell \) for all \( 1 \leq i \leq k \). Find, in terms of \( n \) and \(\ell\), the largest real number \( c \) such that the inequality
\[ 
\sum_{i=1}^{k} \sum_{j=1}^{k} x_{i} x_{j} \frac{|A_{i} \cap A_{j}|^{2}}{|A_{i}| \cdot |A_{j}|} \geq c \left( \sum_{i=1}^{k} x_{i} \right)^{2} 
\]
holds for all positive integers \( k \), all nonnegative real numbers \( x_{1}, x_{2}, \ldots, x_{k} \), and all \(\ell\)-large collections \( A_{1}, A_{2}, \ldots, A_{k} \) of subsets of \(\{1, 2, \ldots, n\} \).

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