#include <bits/stdc++.h>
using namespace std;

const int64_t MOD = 998244353;
void add(int64_t& a, int64_t b){
    a = (a+b) % MOD;
}
void mul(int64_t& a, int64_t b){
    a = a*b % MOD;
}

int64_t power_mod(int64_t num, int64_t power){
    int64_t prod = 1;
    num %= MOD;
    while(power > 0){
        if(power&1) prod = prod * num % MOD;
        num = num * num % MOD;
        power >>= 1;
    }
    return prod;
}

bool online(int x1, int y1, int x2, int y2, int x3, int y3){
    return (y2-y1)*(x3-x1) == (y3-y1)*(x2-x1);
}

int main(){
    int N;
    cin >> N;
    vector<int> X(N), Y(N);
    for(int i=0; i<N; i++) cin >> X[i] >> Y[i];

    int64_t ans = power_mod(2, N);
    add(ans, MOD - 1 - N);

    for(int i=0; i<N; i++) for(int j=i+1; j<N; j++){
        bool bef = false;
        for(int k=0; k<j; k++) if(k!=i && online(X[i], Y[i], X[j], Y[j], X[k], Y[k])) bef = true;
        if(bef) continue;
        int num = 2;
        for(int k=j+1; k<N; k++) num += online(X[i], Y[i], X[j], Y[j], X[k], Y[k]);
        int64_t res = power_mod(2, num) - 1 - num;
        add(ans, MOD - res);
    }
    cout << ans << endl;
}
### Prompt

Create a solution in Cpp for 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 <bits/stdc++.h>
using namespace std;

const int64_t MOD = 998244353;
void add(int64_t& a, int64_t b){
    a = (a+b) % MOD;
}
void mul(int64_t& a, int64_t b){
    a = a*b % MOD;
}

int64_t power_mod(int64_t num, int64_t power){
    int64_t prod = 1;
    num %= MOD;
    while(power > 0){
        if(power&1) prod = prod * num % MOD;
        num = num * num % MOD;
        power >>= 1;
    }
    return prod;
}

bool online(int x1, int y1, int x2, int y2, int x3, int y3){
    return (y2-y1)*(x3-x1) == (y3-y1)*(x2-x1);
}

int main(){
    int N;
    cin >> N;
    vector<int> X(N), Y(N);
    for(int i=0; i<N; i++) cin >> X[i] >> Y[i];

    int64_t ans = power_mod(2, N);
    add(ans, MOD - 1 - N);

    for(int i=0; i<N; i++) for(int j=i+1; j<N; j++){
        bool bef = false;
        for(int k=0; k<j; k++) if(k!=i && online(X[i], Y[i], X[j], Y[j], X[k], Y[k])) bef = true;
        if(bef) continue;
        int num = 2;
        for(int k=j+1; k<N; k++) num += online(X[i], Y[i], X[j], Y[j], X[k], Y[k]);
        int64_t res = power_mod(2, num) - 1 - num;
        add(ans, MOD - res);
    }
    cout << ans << endl;
}
```