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