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 \( (\vec{p}+\vec{q}) \) and \( (\vec{p}-\vec{q}) \) and \( |\vec{r}|=\sqrt{3} \), then \( |\alpha|+|\beta|+|\gamma| \) is equal to __________ .
