Let a line $$L_1$$ pass through the origin and be perpendicular to the lines
$$L_2 : \vec{r} = (3+t)\hat{i} + (2t-1)\hat{j} + (2t+4)\hat{k}$$ and
$$L_3 : \vec{r} = (3+2s)\hat{i} + (3+2s)\hat{j} + (2+s)\hat{k}$$, $$t, s \in \mathbf{R}$$.
If $$(a, b, c)$$, $$a \in \mathbb{Z}$$, is the point on $$L_3$$ at a distance of $$\sqrt{17}$$ from the point of intersection of $$L_1$$ and $$L_2$$, then $$(a + b + c)^2$$ is equal to __________.