
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 a positive integer \( n > 1 \) and a set \( S \) 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 property: for any two distinct elements \( a \) and \( b \) in \( S \), there exists an \( i \) (where \( 1 \leq i \leq k \)) such that \( \left|A_{i} \cap \{a, b\}\right| = 1 \).

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