
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 \( N = \{1, 2, \cdots, n\} \) (where \( n \geq 2 \)), and let \( A_{1}, A_{2}, \cdots, A_{t} \) be subsets of \( N \) forming a family \( F \). The family \( F \) is called distinguishable if for any \( i, j \in N \) with \( i \neq j \), there exists a set \( A_{k} \) such that \( A_{k} \cap \{i, j\} \) contains exactly one element. The family \( F \) is called covering if each element of \( N \) belongs to at least one subset \( A_{i} \in F \). What is the minimum value of \( t \) such that \( F = \{A_{1}, A_{2}, \cdots, A_{i}\} \) is both distinguishable and covering, denoted by \( f(n) \)?

(From the 29th IMO qualifying problem)

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