Let $\vec{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}-\beta \hat{j}+4 \hat{k}$ and $\vec{c}=\hat{i}+2 \hat{j}-2 \hat{k}$ where $\alpha, \beta \in R$, be three vectors. If the projection of $\vec{a}$ on $\vec{c}$ is $\frac{10}{3}$ and $\vec{b} \times \vec{c}=-6 \hat{i}+10 \hat{j}+7 \hat{k}$, then the value of $\alpha+\beta$ equal to :