GMAT Quantitative Section | Re: Solving x + y + z = 4, x^2 + x^2 + z^2 = 4, x^3 + y^3 + z^3 = 4

Knowing that x + y + z = 4
x^2 + x^2 + z^2 = 4 and
x^3 + y^3 + z^3 = 4
Solving the equation with complex numbers

I know that: 4 = x^2 + y^2 + z^2 = (x+y+z)^2 - (x*y + x*z + y*z). So (x*y + x*z + y*z) = 12
4 = x^3 + y^3 + z^3 = (x+y+z^3 - 3*(x^2*y + x^2*z + x*y^2 + x*z^2 + y^2*z + y*z^2) - 6*x*y*z.

Trying to find (x^2*y + x^2*z + x*y^2 + x*z^2 + y^2*z + y*z^2) =
x*(x*y + x*z) + y*(x*y + y*z) + z*(x*z + y*z). Knowing that x+y+z