Let \( \vec{a} = \hat{i} + \alpha \hat{j} + \beta \hat{k} \), \( \alpha, \beta \in \mathbb{R} \). Let a vector \( \vec{b} \) be such that the angle between \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{4} \) and \( |\vec{b}|^2 = 6 \),If \( \vec{a} \times\vec{b} = 3\sqrt{2} \), then the value of \( \left( \alpha^2 + \beta^2 \right) | \vec{a} \times \vec{b} |^2 \) is equal to