(a)[5]
Determine the complex number $z$ that satisfies the equation $z^* + 1 = 2iz$, where $z^*$ represents the complex conjugate of $z$. Write your answer in the form $x + iy$, where $x$ and $y$ are real.
(b(i))[4]
On an Argand sketch, shade the set of points representing complex numbers that satisfy the inequalities $|z + 1 - 3i| \leq 1$ and $\operatorname{Im} z \geq 3$, where $\operatorname{Im} z$ is the imaginary part of $z$.
(b(ii))[2]
Find the difference between the largest and smallest values of $\arg z$ for points in this region.