--description--

Let $n$ be a positive integer. Consider nim positions where:

• There are $n$ non-empty piles.
• Each pile has size less than $2^n$.
• No two piles have the same size.

Let $W(n)$ be the number of winning nim positions satisfying the above conditions (a position is winning if the first player has a winning strategy).

For example, $W(1) = 1$, $W(2) = 6$, $W(3) = 168$, $W(5) = 19\,764\,360$ and $W(100)\bmod 1\,000\,000\,007 = 384\,777\,056$.

Find $W(10\,000\,000)\bmod 1\,000\,000\,007$.

--hints--

nimExtreme() should return 253223948.

assert.strictEqual(nimExtreme(), 253223948);

--seed--

--seed-contents--

function nimExtreme() {

  return true;
}

nimExtreme();

--solutions--

// solution required