#include<bits/stdc++.h>
using namespace std;
int gcd(int a,int b) {return b ? gcd(b,a%b) : abs(a);}
const int N = 205, M = 998244353;
int p2[N];
pair<int,int> a[205];
int main()
{
    int n;
    cin >> n;
    p2[0] = 1;
    for (int i = 1; i <= n; ++i)
        p2[i] = p2[i-1] * 2 % M;
    long long ans = 1;
    for (int i = 0; i < n; ++i) {
        cin >> a[i].first >> a[i].second;
        map<pair<int,int>,int> mp;
        for (int j = 0; j < i; ++j) {
            pair<int,int> v = {a[j].first - a[i].first, a[j].second - a[i].second};
            if (v.first < 0) {
                v.first *= -1;
                v.second *= -1;
            } else if (v.first == 0 && v.second < 0) {
                v.second *= -1;
            }
            int d = gcd(v.first, v.second);
            v.first /= d;
            v.second /= d;
            ++mp[v];
        }
        ++ans;
        for (auto it = begin(mp); it != end(mp); ++it)
            ans += p2[it->second] + M - 1;
    }
    cout << (p2[n] + M - ans%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<bits/stdc++.h>
using namespace std;
int gcd(int a,int b) {return b ? gcd(b,a%b) : abs(a);}
const int N = 205, M = 998244353;
int p2[N];
pair<int,int> a[205];
int main()
{
    int n;
    cin >> n;
    p2[0] = 1;
    for (int i = 1; i <= n; ++i)
        p2[i] = p2[i-1] * 2 % M;
    long long ans = 1;
    for (int i = 0; i < n; ++i) {
        cin >> a[i].first >> a[i].second;
        map<pair<int,int>,int> mp;
        for (int j = 0; j < i; ++j) {
            pair<int,int> v = {a[j].first - a[i].first, a[j].second - a[i].second};
            if (v.first < 0) {
                v.first *= -1;
                v.second *= -1;
            } else if (v.first == 0 && v.second < 0) {
                v.second *= -1;
            }
            int d = gcd(v.first, v.second);
            v.first /= d;
            v.second /= d;
            ++mp[v];
        }
        ++ans;
        for (auto it = begin(mp); it != end(mp); ++it)
            ans += p2[it->second] + M - 1;
    }
    cout << (p2[n] + M - ans%M) % M << endl; 
}

```