The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and $$a _ { n + 1 } = n + 1 + \frac { ( n - 1 ) ! } { a _ { 1 } a _ { 2 } \cdots a _ { n } } \quad ( n \geqq 1 )$$ The following is part of the process of finding the general term $a _ { n }$. For all natural numbers $n$, $$a _ { 1 } a _ { 2 } \cdots a _ { n } a _ { n + 1 } = a _ { 1 } a _ { 2 } \cdots a _ { n } \times ( n + 1 ) + ( n - 1 ) !$$ If $b _ { n } = \frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! }$, then $b _ { 1 } = 1$ and $$b _ { n + 1 } = b _ { n } + ( \text{(가)} )$$ The general term of the sequence $\left\{ b _ { n } \right\}$ is $b _ { n } =$ (나) so $\frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! } =$ (나). $\vdots$ Therefore, $a _ { 1 } = 1$ and $a _ { n } = \frac { ( n - 1 ) ( 2 n - 1 ) } { 2 n - 3 } ( n \geqq 2 )$. When the expression that fits (가) is $f ( n )$ and the expression that fits (나) is $g ( n )$, what is the value of $f ( 13 ) \times g ( 7 )$? [4 points] (1) $\frac { 1 } { 70 }$ (2) $\frac { 1 } { 77 }$ (3) $\frac { 1 } { 84 }$ (4) $\frac { 1 } { 91 }$ (5) $\frac { 1 } { 98 }$
The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and
$$a _ { n + 1 } = n + 1 + \frac { ( n - 1 ) ! } { a _ { 1 } a _ { 2 } \cdots a _ { n } } \quad ( n \geqq 1 )$$
The following is part of the process of finding the general term $a _ { n }$.
For all natural numbers $n$,
$$a _ { 1 } a _ { 2 } \cdots a _ { n } a _ { n + 1 } = a _ { 1 } a _ { 2 } \cdots a _ { n } \times ( n + 1 ) + ( n - 1 ) !$$
If $b _ { n } = \frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! }$, then $b _ { 1 } = 1$ and
$$b _ { n + 1 } = b _ { n } + ( \text{(가)} )$$
The general term of the sequence $\left\{ b _ { n } \right\}$ is\\
$b _ { n } =$ (나) so $\frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! } =$ (나).\\
$\vdots$\\
Therefore, $a _ { 1 } = 1$ and $a _ { n } = \frac { ( n - 1 ) ( 2 n - 1 ) } { 2 n - 3 } ( n \geqq 2 )$.
When the expression that fits (가) is $f ( n )$ and the expression that fits (나) is $g ( n )$, what is the value of $f ( 13 ) \times g ( 7 )$? [4 points]\\
(1) $\frac { 1 } { 70 }$\\
(2) $\frac { 1 } { 77 }$\\
(3) $\frac { 1 } { 84 }$\\
(4) $\frac { 1 } { 91 }$\\
(5) $\frac { 1 } { 98 }$