--description--

The kernel of a polygon is defined by the set of points from which the entire polygon's boundary is visible. We define a polar polygon as a polygon for which the origin is strictly contained inside its kernel.

For this problem, a polygon can have collinear consecutive vertices. However, a polygon still cannot have self-intersection and cannot have zero area.

For example, only the first of the following is a polar polygon (the kernels of the second, third, and fourth do not strictly contain the origin, and the fifth does not have a kernel at all):

Notice that the first polygon has three consecutive collinear vertices.

Let $P(n)$ be the number of polar polygons such that the vertices $(x, y)$ have integer coordinates whose absolute values are not greater than $n$.

Note that polygons should be counted as different if they have different set of edges, even if they enclose the same area. For example, the polygon with vertices [(0,0), (0,3), (1,1), (3,0)] is distinct from the polygon with vertices [(0,0), (0,3), (1,1), (3,0), (1,0)].

For example, $P(1) = 131$, $P(2) = 1\,648\,531$, $P(3) = 1\,099\,461\,296\,175$ and $P(343)\bmod 1\,000\,000\,007 = 937\,293\,740$.

Find $P(7^{13})\bmod 1\,000\,000\,007$.

--hints--

polarPolygons() should return 585965659.

assert.strictEqual(polarPolygons(), 585965659);

--seed--

--seed-contents--

function polarPolygons() {

  return true;
}

polarPolygons();

--solutions--

// solution required