Statement

There are N gnomes, each wearing a hat with a unique integer from 0 to N-1. The gnomes stand in a line in some arbitrary order. Your task is to sort them in increasing order from left to right. In one operation, you can select any subset of gnomes and give them the following instruction: The selected gnomes can only swap positions among themselves, and after rearranging, they must occupy the same set of positions as before. The chosen gnomes do not need to be adjacent. It can be proven that there is always exactly one way to execute this instruction correctly. However, the gnomes are both lazy and stubborn -- they refuse to follow orders more than once. This means that each gnome can be included in at most one operation. Your task: (1) Determine whether it is possible to sort the gnomes. (2) If sorting is possible, compute the minimum number of operations required. (3) Provide a sequence of operations that achieves sorting and has minimum length.