Solution
= Solution
In the ordered <orthonormal basis>
$$
|b\rangle=|\psi_0\rangle|0\rangle,
\qquad
|g\rangle=|\psi_1\rangle|1\rangle
$$
of $\mathcal S$, the <Pauli Z gate> acts as
$$
I_{m-1}\otimes Z
=\begin{pmatrix}1&0\\0&-1\end{pmatrix}
=I_{\mathcal S}-2|g\rangle\langle g|.
$$
It fixes the line $|g\rangle^\perp\cap\mathcal S$ and negates $|g\rangle$, so it is the <reflection in a hyperplane> orthogonal to the good state.