Mathematics Question from JEE Main 2023 · 10 April, Shift 2

Let $\vec{a} = 2\hat{i} + 7\hat{j} - \hat{k}, \, \vec{b} = 3\hat{i} + 5\hat{k}, \, \vec{c} = \hat{i} - \hat{j} + 2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 12$. The value of $\left( -\hat{i} + \hat{j} - \hat{k} \right) \cdot \left( \vec{c} \times \vec{d} \right)$ is: