设 $\alpha=\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right), \beta=\left(\begin{array}{l}1 \\ \frac{1}{2} \\ 0\end{array}\right), \gamma=\left(\begin{array}{l}0 \\ 0 \\ 8\end{array}\right), A=\alpha \beta^T, B=\beta^T \alpha$ ，
其中 $\beta^T$ 是 $\beta$ 的转置，求解方程 $2 B^2 A^2 x=A^4 x+B^4 x+\gamma$.