
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
The problem of partitioning a positive integer is given by:
$$
n = n_1 + n_2 + \cdots + n_k, \quad n_i > 0, \quad 1 \leqslant i \leqslant k
$$

This is called the $k$-partition of the positive integer $n$. If different permutations of $n_i$ are considered the same partition, it is called an unordered partition; otherwise, it is called an ordered partition.

For ordered partitions, there are known to be $\mathrm{C}_{n-1}^{k-1}$ types. Now, we discuss unordered partitions, which we will simply refer to as partitions from here on. Let $B(n)$ denote the total number of partitions of $n$, $B_k(n)$ denote the number of partitions of $n$ where each part is no greater than $k$, and $B_H(n)$ denote the number of partitions of $n$ where each part belongs to $H$. The generating functions corresponding to these are:
1. $\sum_{n=0}^{\infty} B_k(n) x^n = \frac{1}{(1-x)(1-x^2) \cdots (1-x^k)}$;
2. $\sum_{n=0}^{\infty} B_H(n) x^n = \prod_{j \in H} \frac{1}{1-x^j}$;
3. $\sum_{n=0}^{\infty} B(n) x^n = \prod_{j=1}^{\infty} \frac{1}{1-x^j}$.

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