
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 an $n$-element set $X$, a partition of $X$ into subsets $A_{1}, A_{2}, \cdots, A_{m}$ is described by $k_{1}$ single-element subsets, $k_{2}$ two-element subsets, $\cdots, k_{n}$ $n$-element subsets, such that $k_{1} + k_{2} + \cdots + k_{n} = m$ and $1k_{1} + 2k_{2} + \cdots + nk_{n} = n$, where $k_{1}, k_{2}, \cdots, k_{n}$ are all non-negative integers. This partition is denoted as $1^{k_{1}} \cdot 2^{k_{2}} \cdots \cdot n^{k_{n}}$. Determine the number of such partitions.

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