Consider the complex function $M(z) = \frac{mz}{mz - z + 1}$, where $m$ is a complex number such that $|m| = 1$ and $m \neq 1$.

\begin{enumerate}
  \item Find all fixed points of $M(z)$ which satisfy $M(z) = z$.
  \item Express the derivative of $M(z)$ at $z = 0$ by using $m$.
  \item Find $m$ for which the circle $\left| z - \frac{1-i}{2} \right| = \frac{1}{\sqrt{2}}$ on the complex $z$ plane is mapped onto the real axis through $M(z)$.
\end{enumerate}