Try Eliminating

$\begin{aligned} && x-2y = 0 \\ && y+2x = 0 \\ && y= mx - 1 \end{aligned}$

Let $m$ be a real constant such that vertices of the triangle $ABC$ are the intersection point of the equations above.

The equation of the locus of the centroid of triangle $ABC$ is of the form $ax^2 + bxy + cy^2 + dx + ey + f = 0 ,$

where $a,b,c,d,e,f$ are constant coprime integers.

Find $|a+b+c+d+e+f|$ .