We denote by $\mathbb{C}$ the set of complex numbers.\\
In the complex plane equipped with an orthonormal coordinate system $(O; \vec{u}, \vec{v})$ we have placed a point $M$ with affixe $z$ belonging to $\mathbb{C}$, then the point $R$ intersection of the circle with center $O$ passing through $M$ and the half-axis $[O; \vec{u})$.

\textbf{Part A}

\begin{enumerate}
  \item Express the affixe of point $R$ as a function of $z$.
  \item Let the point $M'$ with affixe $z'$ defined by
$$z' = \frac{1}{2}\left(\frac{z + |z|}{2}\right).$$
Reproduce the figure on the answer sheet and construct the point $M'$.
\end{enumerate}

\textbf{Part B}

We define the sequence of complex numbers $(z_n)$ by a first term $z_0$ belonging to $\mathbb{C}$ and, for every natural integer $n$, by the recurrence relation:
$$z_{n+1} = \frac{z_n + |z_n|}{4}$$
The purpose of this part is to study whether the behavior at infinity of the sequence $(|z_n|)$ depends on the choice of $z_0$.

\begin{enumerate}
  \item What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a negative real number?
  \item What can we say about the behavior at infinity of the sequence $(|z_n|)$ when $z_0$ is a positive real number?
  \item We now assume that $z_0$ is not a real number.\\
a. What conjecture can we make about the behavior at infinity of the sequence $(|z_n|)$?\\
b. Prove this conjecture, then conclude.
\end{enumerate}