Let
$M=\begin{bmatrix}sin^{4}\theta &-1-sin^{2}\theta\\ 1+cos^{2}\theta&cos^{4}\theta\end{bmatrix}=\alpha I+\beta M^{-1}$
where $\alpha=\alpha\left(\theta\right)$ and $\beta=\beta\left(\theta\right)$ are real numbers, and $I$ is the 2 x 2 identity matrix. If
$\alpha^*$ is the minimum of the set $\left\{\alpha\left(\theta\right):\theta\,\in[\,0,2\pi)\right\}$ and
$\beta^*$ is the minimum of the set $\left\{\beta\left(\theta\right):\theta\,\in[\,0,2\pi)\right\}$,
then the value of $\alpha^*+\beta^*$ is