For a sequence $\left\{ a _ { n } \right\}$ with first term 1, let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. The following holds:

$$n S _ { n + 1 } = ( n + 2 ) S _ { n } + ( n + 1 ) ^ { 3 } \quad ( n \geq 1 )$$

The following is part of the process of finding the general term of the sequence $\left\{ a _ { n } \right\}$.

Since $S _ { n + 1 } = S _ { n } + a _ { n + 1 }$ for natural numbers $n$,

$$n a _ { n + 1 } = 2 S _ { n } + ( n + 1 ) ^ { 3 } \quad \cdots (\text{ㄱ})$$

For natural numbers $n \geq 2$,

$$( n - 1 ) a _ { n } = 2 S _ { n - 1 } + n ^ { 3 } \quad \cdots (\text{ㄴ})$$

Subtracting (ㄴ) from (ㄱ), we obtain

$$n a _ { n + 1 } = ( n + 1 ) a _ { n } + \text{ (A) }$$

Dividing both sides by $n ( n + 1 )$,

$$\frac { a _ { n + 1 } } { n + 1 } = \frac { a _ { n } } { n } + \frac { \text{ (A) } } { n ( n + 1 ) }$$

Let $b _ { n } = \frac { a _ { n } } { n }$. Then

$$b _ { n + 1 } = b _ { n } + 3 + \text{ (B) } \quad ( n \geq 2 )$$

Therefore

$$b _ { n } = b _ { 2 } + \text{ (C) } \quad ( n \geq 3 )$$

holds.

What are the correct expressions for (A), (B), and (C)?