The director of Hind Circus has decided to add a new performance called the monkey dance to his show. The monkey dance is danced simultaneously by N monkeys. There are N circles drawn on the ground. There are N arrows drawn between the circles in such a way that for each circle, exactly one arrow begins at that circle and exactly one arrow ends at that circle. No arrow can both begin and end at the same circle. When the show begins, each monkey sits on a different circle. At each whistle of the ringmaster, all the monkeys simultaneously jump from one circle to the next, following the arrow leading out of the current circle. This is one step of the dance. The dance ends when all the monkeys have simultaneously returned to the circles where they initially started. The director wishes the dance to last as many steps as possible. This can be achieved by drawing the arrows intelligently. For each of the three values of N given below, what is the maximum number of steps that the monkey dance can be made to last by drawing arrows appropriately? (a) 9 (b) 12 (c) 15
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