Magical Cards

A deck of 2n2n cards is numbered 1,2,,2n1, 2, \cdots, 2n from top to bottom. The top nn cards are removed, kept in order, and put into another pile A. The remaining cards in the original deck are in pile B. Then, the top card is taken from pile B and placed on the table, then the top card from pile A is placed on top of that, then the new top card on pile B is placed on top of that, and so forth until all cards are exhausted. Define a card position mm to be magical if there exists a deck of cards such that the card in position mm retains its original position after the process. As an explicit example, 33 and 66 are magical because in a deck of 88 cards, 33 and 66 are still in the 3rd3rd and 6th6th positions, respectively, counting from the top after the process.

How many integers m<1000m < 1000 are not magical?

Details and Assumptions

A deck of cards must have an even number of cards by definition.

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