Question 4
For a fixed integer $n \ge 2$, determine complex numbers $z$ that satisfy the relations:
$$
a)\ z^n + z^{n-1} + \dots + z^2 + |z| = n;
$$
$$
b)\ |z|^{n-1} + |z|^{n-2} + \dots + |z|^2 + z = n z^n.
$$
$$
a)\ z^n + z^{n-1} + \dots + z^2 + |z| = n;
$$
$$
b)\ |z|^{n-1} + |z|^{n-2} + \dots + |z|^2 + z = n z^n.
$$