Problem 325
Stone Game II
A game is played with two piles of stones and two players.
On each player’s turn, the player may remove a number of stones from the larger pile.
The number of removed 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 $(x_i,y_i)$, $0 < x_i < y_i \le N$.
We can verify that $S(10) = 211$ and $S(10^4) = 230312207313$.
Find $S(10^{16}) \bmod 7^{10}$.
取石子游戏（二）
两名玩家正在用两堆石子玩游戏。
每一轮，玩家可以从较大的堆中取走一定数量的石子。
移走石子的数量必须是较小的堆中石子数量的正整数倍。
例如，用有序对$(6,14)$表示游戏中较小的堆有$6$枚石子，较大的堆有$14$枚石子，先手玩家可以从较大的堆中取走$6$枚或$12$枚石子。
从某一堆中取走所有石头的玩家赢得游戏。
所谓必胜态指先手玩家必胜的状态。例如$(1,5)$、$(2,6)$和$(3,12)$都是必胜态，因为先手玩家可以立即取走第二个堆中的所有石子。
所谓必败态指无论先手玩家如何操作，后手玩家都必胜的状态。例如，$(2,3)$和$(3,4)$都是必败态：先手玩家的任意合法操作都会给后手玩家留下必胜态。
对于所有$0 < x_i < y_i \le N$的必败态$(x_i,y_i)$，记$S(N)$为$(x_i+y_i)$的和。
可以验证$S(10) = 211$，$S(10^4) = 230312207313$。
求$S(10^{16}) \bmod 7^{10}$。
译注：“取石子游戏（一）”参见第260题。