OBM 2018 Problem 2

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$ ; or by $q-1$ ; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$ . The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations.a) Show that if the initial number written is $0$ , then Azambuja cannot reach his goal.b) Find all initial numbers for which Azambuja can achieve his goal.