$(1)$ Find the general solution to the system

$\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 0& 1& 0& 1\\ 2& 1& 2& 1& 2\\ 4& 2& 4& 2& 4\\ 1& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\\ d\\ \\ f\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 0\\ 0\\ 0\end{array}\right]$

or show that there is no solution.