Mathematics Question from JEE Main 2023 · 31 January, Shift 2

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