#include <bits/stdc++.h>
using namespace std;
int n;
const int N = 20000;
const int MOD = 998244353;
int powi(int a, int b)
{
    int c = 1;
    for (; b; b >>= 1, a = 1ll * a * a % MOD)
        if (b & 1) c = 1ll * c * a % MOD;
    return c;
}
map <int, int> M;
int xi[N], yi[N];
int ans;
int main()
{
    cin >> n;
    for (int i = 1; i <= n; ++ i)
        cin >> xi[i] >> yi[i];
    ans = powi(2, n) - n - 1;
    for (int i = 1; i <= n; ++ i)
    {
        M.clear();
        for (int j = i + 1; j <= n; ++ j)
        {
            int d = (1ll * (yi[j] - yi[i]) * powi(xi[j] - xi[i], MOD - 2) % MOD + MOD) % MOD;
            if (xi[j] - xi[i] == 0) d = -1;
            M[d] ++;
        }
        for (map <int, int> :: iterator it = M.begin(); it != M.end(); ++ it)
            ans = (ans - (powi(2, it -> second) - 1)) % MOD;
    }
    ans = (ans + MOD) % MOD;
    cout << ans;
}

### Prompt

Your task is to create a Cpp solution 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 <bits/stdc++.h>
using namespace std;
int n;
const int N = 20000;
const int MOD = 998244353;
int powi(int a, int b)
{
    int c = 1;
    for (; b; b >>= 1, a = 1ll * a * a % MOD)
        if (b & 1) c = 1ll * c * a % MOD;
    return c;
}
map <int, int> M;
int xi[N], yi[N];
int ans;
int main()
{
    cin >> n;
    for (int i = 1; i <= n; ++ i)
        cin >> xi[i] >> yi[i];
    ans = powi(2, n) - n - 1;
    for (int i = 1; i <= n; ++ i)
    {
        M.clear();
        for (int j = i + 1; j <= n; ++ j)
        {
            int d = (1ll * (yi[j] - yi[i]) * powi(xi[j] - xi[i], MOD - 2) % MOD + MOD) % MOD;
            if (xi[j] - xi[i] == 0) d = -1;
            M[d] ++;
        }
        for (map <int, int> :: iterator it = M.begin(); it != M.end(); ++ it)
            ans = (ans - (powi(2, it -> second) - 1)) % MOD;
    }
    ans = (ans + MOD) % MOD;
    cout << ans;
}

```