Problem 325

Stone Game II

A game is played with two piles of stones and two players. At her turn, a player removes a number of stones from the larger pile. The number of stones she removes must be a positive multiple of the number of stones in the smaller pile.

E.g., let the ordered pair(6,14) describe a configuration with 6 stones in the smaller pile and 14 stones in the larger pile, then the first player can remove 6 or 12 stones from the larger pile.

The player taking all the stones from a pile wins the game.

A *winning configuration* is one where the first player can force a
win. For example, (1,5), (2,6) and (3,12) are winning configurations
because the first player can immediately remove all stones in the second
pile.

A *losing configuration* is one where the second player can force a
win, no matter what the first player does. For example, (2,3) and (3,4)
are losing configurations: any legal move leaves a winning
configuration for the second player.

Define S(`N`) as the sum of (`x`_{i}+`y`_{i}) for all losing configurations (`x`_{i},`y`_{i}), 0 `x`_{i}`y`_{i}`N`. We can verify that S(10) = 211 and S(10^{4}) = 230312207313.

Find S(10^{16}) mod 7^{10}.

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These problems are part of
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