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