Let the lines
\( L_1:\ \vec{r}=\lambda(\hat{i}+2\hat{j}+3\hat{k}),\ \lambda\in\mathbb{R} \) and
\( L_2:\ \vec{r}=(\hat{i}+3\hat{j}+\hat{k})+\mu(\hat{i}+\hat{j}+5\hat{k}),\ \mu\in\mathbb{R} \)
Intersect at the point \( S \). If a plane \( ax+by-z+d=0 \) passes through \( S \) and is parallel to both the lines \( L_1 \) and \( L_2 \), then the value of \( a+b+d \) is equal to ________.
