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 stones 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 $S({10}^{16})mod{7}^{10}$.