Let a line $L_1$ pass through the origin and be perpendicular to the lines $$ \begin{aligned} & \mathrm{L}_2: \overrightarrow{\mathrm{r}}=(3+\mathrm{t}) \hat{i}+(2 \mathrm{t}-1) \hat{j}+(2 \mathrm{t}+4) \hat{k} \text { and } \\ & \mathrm{L}_3: \overrightarrow{\mathrm{r}}=(3+2 \mathrm{~s}) \hat{i}+(3+2 \mathrm{~s}) \hat{j}+(2+\mathrm{s}) \hat{k}, \mathrm{t}, \mathrm{~s} \in \mathbf{R} \end{aligned} $$ If $(a, b, c), a \in \mathbf{Z}$, is the point on $\mathrm{L}_3$ at a distance of $\sqrt{17}$ from the point of intersection of $\mathrm{L}_1$ and $\mathrm{L}_2$, then $(\mathrm{a}+\mathrm{b}+\mathrm{c})^2$ is equal to $\_\_\_\_$