2010 AIME I Problem 7

Define an ordered triple (A, B, C) of sets to be \textit{minimally intersecting} if |A \cap B| = |B \cap C| = |C \cap A| = 1 and A \cap B \cap C = \emptyset. For example, (\{1,2\},\{2,3\},\{1,3,4\}) is a minimally intersecting triple. Let N be the number of minimally intersecting ordered triples of sets for which each set is a subset of \{1,2,3,4,5,6,7\}. Find the remainder when N is divided by 1000.

Note: |S| represents the number of elements in the set S.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.