
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 \( P_{m}=\{A_{1}, A_{2}, \cdots, A_{m}\} \) be a partition of an \( n \)-element set \( X \) (i.e., \( A_{1}, A_{2}, \cdots, A_{m} \) partition \( X \)). By splitting one \( A_{i} \) into two sets, we generate a partition \( P_{m+1} \) of \( X \), which consists of \( m+1 \) sets. \( P_{m+1} \) is called a refinement of \( P_{m} \). If 
\[ P_{1}, P_{2}, \cdots, P_{n} \]

are partitions of an \( n \)-element set \( X \), and each is a refinement of the previous one (obviously \( P_{n} \) consists of \( n \) sets, and \( P_{1} \) consists of only one set, \( X \)), this sequence is called a chain of length \( n \). Determine the number of chains of length \( n \).

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