Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$, and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192 \sin ^2 \alpha$ is equal to ________.

<p>$$\begin{aligned} & \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=(2 \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{b}}-3|\mathrm{b}|^2 \\ & |\mathrm{~b}||c| \cos \alpha=-3|\mathrm{~b}|^2 \\ & |\mathrm{c}| \cos \alpha=-12 \text {, as }|\mathrm{b}|=4 \\ & \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=2 \\ & \cos \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3} \\ & |c|^2=|(2 \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})-3 \overrightarrow{\mathrm{b}}|^2 \\ & =64 \times \frac{3}{4}+144=192 \\ & |c|^2 \cos ^2 \alpha=144 \\ & 192 \cos ^2 \alpha=144 \\ & 192 \sin ^2 \alpha=48 \end{aligned}$$</p>