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 \( \dfrac{10}{3} \) and \( \vec{b} \times \vec{c}=-6\hat{i}+10\hat{j}+7\hat{k} \), then the value of \( \alpha+\beta \) is equal to:
