Read it twice

I will share a solution soon, but here are two examples of possible values for $a,b,c$ that satisfy the system of equations above , but $|a|+|b|+|c|$ is not the same , watch out that $a,b,c$ can be complex

if $a,b,c$ are $a=0,b=3,c=3$ these values satisfy the system of equations above and |a|+|b|+|c|=6

if $a,b,c$ are $a=10,b=-2-i\sqrt{45},c=-2+i\sqrt{45}$ these values satisfy the system of equations above and |a|+|b|+|c|=24

$(a+b+c)^2=a^2+b^2+c^2+2(a+b+c)$ $(a+b+c)^2=18+2(9)=36$ $a+b+c=\pm 6$ $|a|+|b|+|c|=6$ only if $a,b,c \ge 0$ or $a,b,c \le 0$ . For example, $a=2,b=7,c=-3\implies a+b+c=6$ but $|a|+|b|+|c|=12$