Let $\quad \bar{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=3 \hat{i}-\hat{j}+\beta \hat{k} \quad$ and $\bar{c}=\hat{i}+2 \hat{j}-2 \hat{k}$ where $\alpha, \beta \in \mathbb{R}$, be three vectors. If the projection of $\bar{a}$ on $\bar{c}$ is $\frac{10}{3}$ and $\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}$, then the value of $(\alpha+\beta)$ is equal to