A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = a _ { 2 } = 1$, and when $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + ( 2 n - 1 ) S _ { n } \quad ( n \geq 2 )$$ The following is the process of finding the general term $a _ { n }$.
Since $a _ { n + 1 } = S _ { n + 1 } - S _ { n }$, from the given equation we have $$S _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + 2 n S _ { n } \quad ( n \geq 2 )$$ Dividing both sides by $S _ { n }$, $$\frac { S _ { n + 1 } } { S _ { n } } = \frac { S _ { n } } { S _ { n - 1 } } + 2 n$$ Let $b _ { n } = \frac { S _ { n + 1 } } { S _ { n } }$. Then $b _ { 1 } = 2$ and $$b _ { n } = b _ { n - 1 } + 2 n \quad ( n \geq 2 )$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \text { (가) } \times ( n + 1 ) \quad ( n \geq 1 )$$ Therefore, $$S _ { n } = ( \text{가} ) \times \{ ( n - 1 ) ! \} ^ { 2 } \quad ( n \geq 1 )$$ Thus $a _ { 1 } = 1$, and for $n \geq 2$, $$\begin{aligned} a _ { n } & = S _ { n } - S _ { n - 1 } \\ & = \text { (나) } \times \{ ( n - 2 ) ! \} ^ { 2 } \end{aligned}$$ When the expressions for (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $f ( 10 ) + g ( 6 )$? [4 points]
(1) 110
(2) 125
(3) 140
(4) 155
(5) 170