Let $S = \left\{ \sin ^ { 2 } 2 \theta : \left( \sin ^ { 4 } \theta + \cos ^ { 4 } \theta \right) x ^ { 2 } + ( \sin 2 \theta ) x + \left( \sin ^ { 6 } \theta + \cos ^ { 6 } \theta \right) = 0 \right.$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3 \left( ( \alpha - 2 ) ^ { 2 } + ( \beta - 1 ) ^ { 2 } \right)$ equals $\_\_\_\_$