Let $\vec a=2\hat i+\alpha\hat j+\hat k,\ \vec b=-\hat i+\hat k,\ \vec c=\beta\hat j-\hat k$, where $\alpha,\beta$ are integers and $\alpha\beta=-6$. Let the values of the ordered pair $(\alpha,\beta)$ for which the area of the parallelogram whose diagonals are $\vec a+\vec b$ and $\vec b+\vec c$ is $\dfrac{\sqrt{21}}{2}$ be $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$. Then $\alpha_1^{,2}+\beta_1^{,2}-\alpha_2\beta_2$ is equal to: