Let $\vec{p}=2 \hat{i}+3 \hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2 \hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})$ is perpendicular to each of the vectors $(\bar{p}+\bar{q})$ and $(\bar{p}-\bar{q})$, and $|\bar{r}|=\sqrt{3}$, then $|\alpha|+|\beta|+|r|$ is equal to $\_\_\_\_$ .