BALKAN-SL 2020 N2

A number of $N$ children are at a party and they sit in a circle to play a game of Pass and Parcel. Because the host has no other form of entertainment, the parcel has infinitely many layers. On turn $i$ , starting with $i=1$ , the following two things happen in order:$(1)$ The parcel is passed $i^2$ positions clockwise; and**$(2)$** The child currently holding the parcel unwraps a layer and claims the prize inside. For what values of $N$ will every chidren receive a prize?

$Patrick \ Winter United \ Kingdom$