We are distributing some identical candies into three non-empty piles such that the number of candies in each pile is different. How many candies do we have if the number of different possible groupings is exactly one more than the number of candies?

We have a certain number of identical candies, and we need to distribute them into three non-empty piles in such a way that the number of candies in each pile is different. If the number of different possible groupings is exactly one more than the number of candies, how many candies do we have?