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

using ll = long long;

constexpr ll mod = 998244353;

ll modpow(ll x, ll n) {
    ll res = 1;
    while(n > 0) {
        if(n & 1) (res *= x) %= mod;
        (x *= x) %= mod;
        n >>= 1;
    }
    return res;
}

int main() {
    int n; cin >> n;
    vector<int> x(n), y(n);
    for(int i = 0; i < n; ++i) {
        cin >> x[i] >> y[i];
    }

    ll ans = (modpow(2, n) - n - 1 + mod) % mod;
    set<vector<int>> vss;
    for(int i = 0; i < n; ++i) {
        for(int j = i + 1; j < n; ++j) {
            vector<int> vs;
            for(int k = 0; k < n; ++k) {
                const int dx1 = x[j] - x[i], dx2 = x[k] - x[i];
                const int dy1 = y[j] - y[i], dy2 = y[k] - y[i];
                if(dx1 * dy2 - dx2 * dy1 == 0) {
                    vs.push_back(k);
                }
            }
            if(vss.count(vs)) continue;
            ans = (ans - modpow(2, vs.size()) + vs.size() + 1 + mod) % mod;
            vss.insert(move(vs));
        }
    }

    cout << ans << endl;
}
### Prompt

In CPP, your task is to solve 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;

using ll = long long;

constexpr ll mod = 998244353;

ll modpow(ll x, ll n) {
    ll res = 1;
    while(n > 0) {
        if(n & 1) (res *= x) %= mod;
        (x *= x) %= mod;
        n >>= 1;
    }
    return res;
}

int main() {
    int n; cin >> n;
    vector<int> x(n), y(n);
    for(int i = 0; i < n; ++i) {
        cin >> x[i] >> y[i];
    }

    ll ans = (modpow(2, n) - n - 1 + mod) % mod;
    set<vector<int>> vss;
    for(int i = 0; i < n; ++i) {
        for(int j = i + 1; j < n; ++j) {
            vector<int> vs;
            for(int k = 0; k < n; ++k) {
                const int dx1 = x[j] - x[i], dx2 = x[k] - x[i];
                const int dy1 = y[j] - y[i], dy2 = y[k] - y[i];
                if(dx1 * dy2 - dx2 * dy1 == 0) {
                    vs.push_back(k);
                }
            }
            if(vss.count(vs)) continue;
            ans = (ans - modpow(2, vs.size()) + vs.size() + 1 + mod) % mod;
            vss.insert(move(vs));
        }
    }

    cout << ans << endl;
}
```