APMO 2025 Problem 4

Let $n \geq 3$ be an integer. There are $n$ cells on a circle, and each cell is assigned either 0 or 1. There is a rooster on one of these cells, and it repeats the following operations:

If the rooster is on a cell assigned 0, it changes the assigned number to 1 and moves to the next cell counterclockwise. If the rooster is on a cell assigned 1, it changes the assigned number to 0 and moves to the cell after the next cell counterclockwise.

Prove that the following statement holds true after sufficiently many operations:

If the rooster is on a cell $C$ , then the rooster would go around the circle exactly three times, stopping again at $C$ . Moreover, every cell would be assigned the same number as it was assigned right before the rooster went around the circle 3 times.