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Equal votes in a village election
High school · ●●○○○ 20 pts
A small village has people. During their yearly elections, every group of three people comes up to a stage and, as a group, casts a single vote for someone in the village to be the new leader. After every possible group of three people has voted, it turns out that every person in the village received exactly the same number of votes. If , how many possible values of are there?