Willy's Extreme New Game

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Willy has just created an extreme version of his new game. Please refer to the first paragraph of this problem for information about rules and how to play.

In this extreme version, Willy has changed the rules so that each player must count \(2^{n}\) consecutive natural numbers, where \(1 \le n \le 10\).

How many different games are possible if the first player to count the number \(500,000\) wins? Express your answer as the sum of the digits of this number.

Note that a game is different than another game if, for all \(1 \le n \le 10\), the number of turns of length \(2^{n}\) in each game are different.


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