#include <iostream>
using namespace std;

long long modexp(int x, long long e, int m) {
    long long ans = 1, p = x % m;
    while (e > 0) {
        if (e % 2 != 0) ans = (ans * p) % m;
        p = (p * p) % m;
        e >>= 1;
    }
    return ans;
}

const int M = 998244353;

struct P { int x, y; };

bool colinear(P a, P b, P c) {
    int x1 = a.x - c.x, y1 = a.y - c.y,
        x2 = b.x - c.x, y2 = b.y - c.y;
    return x1 * y2 == x2 * y1;
}

int main() {
    int n; cin >> n;
    P p[n]; for (int i = 0; i < n; i++) cin >> p[i].x >> p[i].y;
    long long ans = modexp(2, n, M) - 1 - n - n * (n - 1) / 2;
    for (int i = 0; i < n; i++)
        for (int j = i+1; j < n; j++) {
            int count = 0;
            for (int k = j+1; k < n; k++)
                if (colinear(p[i], p[j], p[k])) count++;
            if (count > 0)
                ans -= modexp(2, count, M) - 1;
        }
    cout << (ans % M + M) % M << endl;
}

### Prompt

Please create a solution in Cpp to the following problem:

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

### Response

```cpp
#include <iostream>
using namespace std;

long long modexp(int x, long long e, int m) {
    long long ans = 1, p = x % m;
    while (e > 0) {
        if (e % 2 != 0) ans = (ans * p) % m;
        p = (p * p) % m;
        e >>= 1;
    }
    return ans;
}

const int M = 998244353;

struct P { int x, y; };

bool colinear(P a, P b, P c) {
    int x1 = a.x - c.x, y1 = a.y - c.y,
        x2 = b.x - c.x, y2 = b.y - c.y;
    return x1 * y2 == x2 * y1;
}

int main() {
    int n; cin >> n;
    P p[n]; for (int i = 0; i < n; i++) cin >> p[i].x >> p[i].y;
    long long ans = modexp(2, n, M) - 1 - n - n * (n - 1) / 2;
    for (int i = 0; i < n; i++)
        for (int j = i+1; j < n; j++) {
            int count = 0;
            for (int k = j+1; k < n; k++)
                if (colinear(p[i], p[j], p[k])) count++;
            if (count > 0)
                ans -= modexp(2, count, M) - 1;
        }
    cout << (ans % M + M) % M << endl;
}

```