Let the lines L_1: r =λ( i +2 j +3 k ), λ ∈ R; L_2: r =( i +3 j + k )+μ( i + j +5 k ); μ ∈ R, intersect at the point S. If a plane a x+ b y-z+ d =0…

Let the lines $$\begin{aligned} & \mathrm{L}_1: \overrightarrow{\mathbf{r}}=\lambda(\hat{i}+2 \hat{j}+3 \hat{k}), \lambda \in \mathbf{R} \\ & \mathrm{L}_2: \overrightarrow{\mathbf{r}}=(\hat{i}+3 \hat{j}+\hat{k})+\mu(\hat{i}+\hat{j}+5 \hat{k}) ; \mu \in \mathbf{R}, \end{aligned} $$ intersect at the point S. If a plane $\displaystyle \mathrm{a} x+\mathrm{b} y-z+\mathrm{d}=0$ passes through S and is parallel to both the lines $\displaystyle \mathrm{L}_1$ and $\displaystyle \mathrm{L}_2$, then the value of $\displaystyle \mathrm{a}+\mathrm{b}+\mathrm{d}$ is equal to \_\_\_\_.