Given a sequence which contains the numbers 1 through 8, a move consists of moving one of the numbers to the front of the sequence. For example, for the sequence $(1, 5, 8, 3, 6, 2, 7, 4),$ we can move the number $6$ to the front, which results in the sequence $(6, 1, 5, 8, 3, 2, 7, 4).$ (We can also choose to move the first number to the front, which does not change the sequence.)

We want to turn a sequence into the sequence (1, 2, 3, 4, 5, 6, 7, 8) with a series of moves. For how many starting sequences does this take a a minimum of two moves?

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