--description--

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 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 $S({10}^4) = 230\,312\,207\,313$.

Find $S({10}^{16})\bmod 7^{10}$.

--hints--

stoneGameTwo() should return 54672965.

assert.strictEqual(stoneGameTwo(), 54672965);

--seed--

--seed-contents--

function stoneGameTwo() {

  return true;
}

stoneGameTwo();

--solutions--

// solution required