AMC 10 2022b Problem 18

Consider systems of three linear equations with unknowns $x$, $y$, and $z$, \begin{align*} a_1 x + b_1 y + c_1 z & = 0 \ a_2 x + b_2 y + c_2 z & = 0 \ a_3 x + b_3 y + c_3 z & = 0 \end{align*} where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x=y=z=0$. For example, one such system is $$\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}$$ with a nonzero solution of $\{x,y,z\} = \{1, -1, 1\}$. How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.) $\textbf{(A)}\ 302 \textbf{(B)}\ 338 \textbf{(C)}\ 340 \textbf{(D)}\ 343 \textbf{(E)}\ 344$