smc

Notice if $a$, $b$, and $c$ are $0$, then we can find $x$, $y$, and $z$ to disprove $\text{I}$ and $\text{II}$. For example, if $(a,b,c,x,y,z)=(0,0,0,-1,-1,-1)$, then $\text{I}$ and $\text{II}$ are disproved. If $(a,b,c,x,y,z)=(0,1,2,-1,-1,1)$, then $\text{III}$ is disproved. Therefore the answer is $\boxed{\text{None are satisfied.}}$.