For two real numbers $a$ and $k$, two functions $f ( x )$ and $g ( x )$ are defined as $$\begin{aligned}
& f ( x ) = \begin{cases} 0 & ( x \leq a ) \\
( x - 1 ) ^ { 2 } ( 2 x + 1 ) & ( x > a ) \end{cases} \\
& g ( x ) = \begin{cases} 0 & ( x \leq k ) \\
12 ( x - k ) & ( x > k ) \end{cases}
\end{aligned}$$ and satisfy the following conditions. (가) The function $f ( x )$ is differentiable on the entire set of real numbers. (나) For all real numbers $x$, $f ( x ) \geq g ( x )$. When the minimum value of $k$ is $\frac { q } { p }$, find the value of $a + p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
For two real numbers $a$ and $k$, two functions $f ( x )$ and $g ( x )$ are defined as
$$\begin{aligned}
& f ( x ) = \begin{cases} 0 & ( x \leq a ) \\
( x - 1 ) ^ { 2 } ( 2 x + 1 ) & ( x > a ) \end{cases} \\
& g ( x ) = \begin{cases} 0 & ( x \leq k ) \\
12 ( x - k ) & ( x > k ) \end{cases}
\end{aligned}$$
and satisfy the following conditions.\\
(가) The function $f ( x )$ is differentiable on the entire set of real numbers.\\
(나) For all real numbers $x$, $f ( x ) \geq g ( x )$.\\
When the minimum value of $k$ is $\frac { q } { p }$, find the value of $a + p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]