`z+y+x=1`
`⇔(x+y+z)^3=1`
`⇔x^3+y^3+z^3+3(x+y)(y+z)(x+z)=1`
`⇔3(x+y)(y+z)(x+z)=0`
`⇔(x+y)(y+z)(x+z)=0`
⇔\(\left[ \begin{array}{l}x+y=0\\y+z=0\\x+z=0\end{array} \right.\)
⇔\(\left[ \begin{array}{l}z=1\\x=1\\y=1\end{array} \right.\)
\left.\begin{matrix} x=1\\y=1\\ z=1 \end{matrix}\right\}`⇒A=0+0+1=1`