Mathematics Question from JEE Main 2023 · 1 February, Shift 1

Let $\vec{v} = \alpha \hat{i} + 2 \hat{j} - 3 \hat{k}, \quad \vec{w} = 2\alpha \hat{i} + \hat{j} - \hat{k}, \quad \text{and } \hat{u} \text{ be a vector such that } |\hat{u}| = \alpha > 0.$ If the minimum value of the scalar triple product $[\vec{u} \quad \vec{v} \quad \vec{w}]$ is $-\alpha \sqrt{3401},$ and $|\hat{u} \cdot \hat{i}|^2 = \frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m + n$ is equal to: