Let a line L_1 pass through the origin and be perpendicular to the lines L_2: r =(3+ t ) i +(2 t -1) j +(2 t +4) k and L_3: r =(3+2 s ) i +(3+2 s ) j…

Let a line $\displaystyle \mathrm{L}_1$ 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 $\displaystyle (\mathrm{a}, \mathrm{b}, \mathrm{c}), \mathrm{a} \in \mathbf{Z}$, is the point on $\displaystyle \mathrm{L}_3$ at a distance of $\displaystyle \sqrt{17}$ from the point of intersection of $\displaystyle \mathrm{L}_1$ and $\displaystyle \mathrm{L}_2$, then $\displaystyle (\mathrm{a}+\mathrm{b}+\mathrm{c})^2$ is equal to $\displaystyle \_\_\_\_$.