
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 \equal{} \{1,2 \ldots, n\}, n \geq 2.$  A collection  $ F \equal{} \{A_1, \ldots, A_t\}$  of subsets  $ A_i \subseteq N,$   $ i \equal{} 1, \ldots, t,$  is said to be separating, if for every pair  $ \{x,y\} \subseteq N,$  there is a set  $ A_i \in F$  so that  $ A_i \cap \{x,y\}$  contains just one element.  $ 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,$  so there is a set  $ F \equal{} \{A_1, \ldots, A_t\}$  which is simultaneously separating and covering?

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