Eccoci.

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, $P(1) = 131$, $P(2) = 1648531$, $P(3) = 1099461296175$ and $P(343) \bmod 1\,000\,000\,007 = 937293740$.

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