At PuzzlingCorp, all employees have a break at 10:00am sharp every day, and immediately proceed to get either plain water or a diet coke.
The people at PuzzlingCorp are also highly susceptible to peer pressure. The way this works is as follows: each one of the employees has a set of other employees that that employee views as her friends. (Friendship is a symmetric relationship.) At 10:00am, when employees drink their morning drinks, each one of them observes the water/diet coke choice of the rest of her peer group. If, in the group, there is a majority for water, the next day that employee will get water for herself. If there is a majority for diet coke, she will go with diet coke. If there is a tie, the employee will simply drink the following day whatever she had to drink on the previous day.
Question. It is clear that ultimately the drinking patterns of PuzzlingCorp's employees will settle on a cycle. What are possible lengths of this cycle?
(Hint: only two lengths are at all possible.)