
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 \( n (\geqslant 2) \) be a positive integer, and \( S \) 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 \( a \) and \( b \) of \( S \), there exists \( j \in \{1, 2, \cdots, k\} \) such that the intersection \( A_{j} \cap \{a, b\} \) is a singleton.

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