[Calculus] The position $( x , y )$ of a point P moving on the coordinate plane at time $t$ is given by
$$\left\{ \begin{array} { l }
x = 4 ( \cos t + \sin t ) \\
y = \cos 2 t
\end{array} \quad ( 0 \leqq t \leqq 2 \pi ) \right.$$
When the distance traveled by point P from $t = 0$ to $t = 2 \pi$ is $a \pi$, find the value of $a ^ { 2 }$. [4 points]