
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 \( n \geq 2 \), let the set \( N = \{1, 2, \ldots, n\} \). A family of subsets \( \mathcal{F} = \{A_1, A_2, \ldots, A_t\} \) of \( N \) is called "separable" if for every pair of distinct elements \( x, y \in N \), there exists a subset \( A_i \in \mathcal{F} \) such that \( |A_i \cap \{x, y\}| = 1 \). The family \( \mathcal{F} \) is called "covering" if every element of \( N \) belongs to at least one \( A_i \in \mathcal{F} \). Determine the minimum value of \( t \) such that there exists a family \( \mathcal{F} = \{A_1, A_2, \ldots, A_t\} \) which is both separable and covering.

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