
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 $n (\geq 2)$ be a positive integer, and let $S$ be a set with $n$ elements. Find the smallest positive integer $k$, such that there exists a collection of subsets $A_{1}, A_{2}, \cdots, A_{k}$ of $S$ with the following property: for any two distinct elements $a, b$ of $S$, there exists some $j \in \{1, 2, \cdots, k\}$ such that $A_{j} \cap \{a, b\}$ is a singleton set.

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