The system of equations $x+2y+z=0,x-z=k,x+y=0$ has infinitely many solutions. If $\text{adj}(A)=\left[\begin{array}{ccc}1& 1& -2\\ -1& -1& 2\\ 1& 1& -2\end{array}\right]$, then the possible value of $k$ is:

1. A
   $k=\pm 1$
2. B
   $k=0$
3. C
   $k$ can have any real value
4. D
   None