For a cubic function $f ( x )$ with leading coefficient $6 \pi$, the function $g ( x ) = \frac { 1 } { 2 + \sin ( f ( x ) ) }$ has a local maximum or minimum at $x = \alpha$, and when all $\alpha \geq 0$ are listed in increasing order as $\alpha _ { 1 }$, $\alpha _ { 2 } , \alpha _ { 3 } , \alpha _ { 4 } , \alpha _ { 5 } , \cdots$, the function $g ( x )$ satisfies the following conditions.\\
(가) $\alpha _ { 1 } = 0$ and $g \left( \alpha _ { 1 } \right) = \frac { 2 } { 5 }$.\\
(나) $\frac { 1 } { g \left( \alpha _ { 5 } \right) } = \frac { 1 } { g \left( \alpha _ { 2 } \right) } + \frac { 1 } { 2 }$\\
When $g ^ { \prime } \left( - \frac { 1 } { 2 } \right) = a \pi$, find the value of $a ^ { 2 }$.\\
(Here, $0 < f ( 0 ) < \frac { \pi } { 2 }$.) [4 points]