#include <iostream>
#include <algorithm>
#include <vector>
#include <string>
#include <cmath>
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
const int mod = 1000000007;

int main() {
    ios::sync_with_stdio(false);
    int n, x[213] = {}, y[213] = {}, p[213] = {1}, z = 0;
    bool b[213][213] = {};
    cin >> n;
    for (int i = 1; i <= n; i++) p[i] = p[i - 1] * 2 % 998244353;
    for (int i = 1; i <= n; i++) cin >> x[i] >> y[i];
    for (int i = 1; i <= n; i++) for (int j = i + 1; j <= n; j++) if (!b[i][j]) {
        vector<int> c = {i, j};
        int w = 2;
        for (int k = j + 1; k <= n; k++) if ((x[i] - x[j]) * (y[i] - y[k]) == (y[i] - y[j]) * (x[i] - x[k])) w++, c.push_back(k);
        for (int k : c) for (int l : c) b[k][l] = 1;
        z = (z + p[w] - 1 - w) % 998244353;
    }
    cout << (p[n] - z - n + 998244352) % 998244353;
}

### 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>
#include <algorithm>
#include <vector>
#include <string>
#include <cmath>
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
const int mod = 1000000007;

int main() {
    ios::sync_with_stdio(false);
    int n, x[213] = {}, y[213] = {}, p[213] = {1}, z = 0;
    bool b[213][213] = {};
    cin >> n;
    for (int i = 1; i <= n; i++) p[i] = p[i - 1] * 2 % 998244353;
    for (int i = 1; i <= n; i++) cin >> x[i] >> y[i];
    for (int i = 1; i <= n; i++) for (int j = i + 1; j <= n; j++) if (!b[i][j]) {
        vector<int> c = {i, j};
        int w = 2;
        for (int k = j + 1; k <= n; k++) if ((x[i] - x[j]) * (y[i] - y[k]) == (y[i] - y[j]) * (x[i] - x[k])) w++, c.push_back(k);
        for (int k : c) for (int l : c) b[k][l] = 1;
        z = (z + p[w] - 1 - w) % 998244353;
    }
    cout << (p[n] - z - n + 998244352) % 998244353;
}

```