p03615 AtCoder Regular Contest 082 - ConvexScore
You are given N points (x_i,y_i) located on a two-dimensional plane. Consider a subset S of the N points that forms a convex polygon. Here, we say a set of points S forms a convex polygon when there exists a convex polygon with a positive area that has the same set of vertices as S. All the interior angles of the polygon must be strictly less than 180°.

cddb0c267926c2add885ca153c47ad8a.png

For example, in the figure above, {A,C,E} and {B,D,E} form convex polygons; {A,C,D,E}, {A,B,C,E}, {A,B,C}, {D,E} and {} do not.

For a given set S, let n be the number of the points among the N points that are inside the convex hull of S (including the boundary and vertices). Then, we will define the score of S as 2^{n-|S|}.

Compute the scores of all possible sets S that form convex polygons, and find the sum of all those scores.

However, since the sum can be extremely large, print the sum modulo 998244353.

Constraints

* 1≤N≤200
* 0≤x_i,y_i<10^4 (1≤i≤N)
* If i≠j, x_i≠x_j or y_i≠y_j.
* x_i and y_i are integers.

Input

The input is given from Standard Input in the following format:


N
x_1 y_1
x_2 y_2
:
x_N y_N


Output

Print the sum of all the scores modulo 998244353.

Examples

Input

4
0 0
0 1
1 0
1 1


Output

5


Input

5
0 0
0 1
0 2
0 3
1 1


Output

11


Input

1
3141 2718


Output

0
```python
import sys
from collections import defaultdict

# sys.stdin = open('e1.in')


def read_int_list():
    return list(map(int, input().split()))


def read_int():
    return int(input())


def read_str_list():
    return input().split()


def read_str():
    return input()


def gcd(a, b):
    if b == 0:
        return a
    return gcd(b, a % b)


def solve():
    def collinear(x0, y0, x1, y1):
        return x0 * y1 == x1 * y0

    def aligned(i, j, k):
        return collinear(x[j] - x[i], y[j] - y[i], x[k] - x[i], y[k] - y[i])

    n = read_int()
    mod = 998244353
    res = pow(2, n, mod) - n - 1
    x, y = zip(*[read_int_list() for i in range(n)])
    lines = defaultdict(set)
    for i in range(n):
        for j in range(i + 1, n):
            a = y[i] - y[j]
            b = x[j] - x[i]
            g = gcd(a, b)
            a //= g
            b //= g
            if a < 0 or (a == 0 and b < 0):
                a, b = -a, -b
            c = -(a * x[i] + b * y[i])
            lines[(a, b, c)].add(i)
            lines[(a, b, c)].add(j)
    for k, v in lines.items():
        m = len(v)
        res -= pow(2, m, mod) - m - 1
    res %= mod
    return res


def main():
    res = solve()
    print(res)


if __name__ == '__main__':
    main()

```