
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
Suppose \( N = \{1, 2, \cdots, n\} \) with \( n \geq 2 \). The subsets \( A_i (i = 1, 2, \cdots, t) \) form a collection \( F = \{A_1, A_2, \cdots, A_t\} \). If for every pair of elements \( x, y \in N \), there exists a set \( A_i \in F \) such that \( A_i \cap \{x, y\} \) contains exactly one element, then \( F \) is called separable. If every element of \( N \) belongs to at least one set \( A_i \in F \), then \( F \) is called covering. What is the minimum value \( t = f(n) \) for which there exists an \( F = \{A_1, A_2, \cdots, A_t\} \) that is both separable and covering?

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