The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$.  Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$.  Find the real part of the sum of the coefficients of $g(x)$.

Let's think step by step and output the final answer within \boxed{}.
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