Subpolygon

This problem has a non-standard time limit of 10 seconds.

Let P be a (strictly) convex polygon, let V be the set of vertices of P, and let W be any subset of V of size 3 or more. Then, the convex hull of W is called a subpolygon of P.

For instance, let (0,0), (2,0), (3,3), (0,2) be the vertices of a convex polygon P. This polygon has five subpolygons: the polygon itself and four triangles.

You are given a special convex polygon P on 4n vertices, all having integer coordinates. Calculate the sum of areas of all subpolygons of P. Express the answer as a fraction A/B, where gcd(A,B) = 1. Let B^-1 be the multiplicative inverse of B modulo 998244353. Return A*B^-1 modulo 998244353.

The procedure for generating the input polygon P is as follows. You are given an int n. Consider the square with vertices (n,0), (0,n), (-n,0), (0,-n). On the boundary of this square there are exactly 4n points with integer coordinates. (This includes all four corners of the square.) Start at an arbitrary one of these 4n points and label them T[0] through T[4n-1] in counter-clockwise order.

Let P[0] be an arbitrary point in the plane. For all i from 1 to 4n, set P[i] = P[i-1] + T[i-1]. I.e., if P[i-1] has coordinates (x1,y1) and T[i-1] has coordinates (x2,y2), the new point P[i] has coordinates (x1+x2,y1+y2).

The polygon P you should process is the polygon with vertices at P[0], P[1], ..., P[4n-1]. Note that P is always a simple convex polygon, and that the answer you should return does not depend on the choices you made during the construction. Also note that the above process sets P[4n] to the same point as P[0].