
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 \( n \geqslant 2 \), divide the set \(\{1,2, \cdots, n\}\) into \( k \) non-empty subsets \( A_{1}, A_{2}, \cdots, A_{k} \) (subsets are unordered), such that
\[ A_{1} \cup A_{2} \cup \cdots \cup A_{k} = \{1,2, \cdots, n\}, \]
and for any \( 1 \leqslant i < j \leqslant k \), \( A_{i} \cap A_{j} \neq \varnothing \). This unordered grouping \(\left( A_{1}, A_{2}, \cdots, A_{k} \right) \) of the set \(\{1,2, \cdots, n\}\) is called an unordered \( k \) partition.

Calculate the number of all distinct unordered \( k \) partitions of the set \(\{1,2, \cdots, n\}\), denoted as \( S(n, k) \) (known as the Stirling number of the second kind).

The recursion relation for \( S(n, k) \) is given by
\[ S(n, k) = k S(n-1, k) + S(n-1, k-1) \quad (k \geqslant 2). \]

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