The point of intersection $$C$$ of the plane $$8x + y + 2z = 0$$ and the line joining the points $$A(-3, -6, 1)$$ and $$B(2, 4, -3)$$ divides the line segment $$AB$$ internally in the ratio $$k : 1$$. If $$a, b, c$$ ($$|a|, |b|, |c|$$ are coprime) are the direction ratios of the perpendicular from the point $$C$$ on the line $$\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}$$, then $$|a + b + c|$$ is equal to _____.

Plane : $$8x + y + 2z = 0$$

Given line AB : $$\frac{x-2}{5} = \frac{y-4}{10} = \frac{z+3}{-4} = \lambda$$

Any point on line: $$(5\lambda + 2, 10\lambda + 4, -4\lambda - 3)$$

Point of intersection of line and plane:

$$8(5\lambda + 2) + 10\lambda + 4 - 8\lambda - 6 = 0$$

$$\lambda = -\frac{1}{3}$$

$$C\left(\frac{1}{3}, \frac{2}{3}, -\frac{5}{3}\right)$$

$$L : \frac{x-1}{-1} = \frac{y+4}{2} = \frac{z+2}{3} = \mu$$

$$D(-\mu+1, 2\mu-4, 3\mu-2)$$

$$\overline{\text{CD}} = \left(-\mu + \frac{2}{3}\right)\hat{i} + \left(2\mu - \frac{14}{3}\right)\hat{j} + \left(3\mu - \frac{1}{3}\right)\hat{k}$$

$$\left(-\mu + \frac{2}{3}\right)(-1) + \left(2\mu - \frac{14}{3}\right)2 + \left(3\mu - \frac{1}{3}\right)3 = 0$$

$$\mu = \frac{11}{14}$$

$$\overline{\text{CD}} = \frac{-5}{42}, \frac{-130}{42}, \frac{85}{42}$$

Direction ratios $$\rightarrow (-1, -26, 17)$$ $$\vert{}a + b + c\vert{} = 10$$