Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be three vectors such that $\left|\overrightarrow{a}\right|=\sqrt{31},4\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=2$ and $2\left(\overrightarrow{a}\times \overrightarrow{b}\right)=3\left(\overrightarrow{c}\times \overrightarrow{a}\right)$. If the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{2\pi }{3}$, then ${\left(\frac{\overrightarrow{a}\times \overrightarrow{c}}{\overrightarrow{a}·\overrightarrow{b}}\right)}^2$ is equal to _____ .