
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 \( N = \{1, 2, \ldots, n\} \) where \( n \geq 2 \). A collection \( F = \{A_1, \ldots, A_t\} \) of subsets \( A_i \subseteq N \), for \( i = 1, \ldots, t \), is said to be separating if for every pair \(\{x, y\} \subseteq N\), there is a set \( A_i \in F \) such that \( A_i \cap \{x, y\} \) contains just one element. A collection \( F \) is said to be covering if every element of \( N \) is contained in at least one set \( A_i \in F \). What is the smallest value \( f(n) \) of \( t \) such that there is a set \( F = \{A_1, \ldots, A_t\} \) that is simultaneously separating and covering?

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