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

#define rep(i, N) for (int i = 0; i < N; i++)
#define pb push_back
typedef long long ll;
typedef pair<int, int> i_i;
typedef pair<ll, int> ll_i;
const int MOD = 998244353;
const int INF = INT_MAX / 2;
struct edge { int v, w; };
#define double long double

double d(int x1, int y1, int x2, int y2) {
	double dx = x1 - x2, dy = y1 - y2;
	return sqrt(dx * dx + dy * dy);
}

int main() {
	int N; cin >> N;
	vector<int> x(N), y(N);
	rep(i, N) cin >> x[i] >> y[i];
	int ans = 1;
	rep(t, N) ans = ans * 2 % MOD;
	rep(i, N) rep(j, i) {
		int n = 0;
		rep(k, N) if (k != i && k != j) {
			double d1 = d(x[k], y[k], x[i], y[i]);
			double d2 = d(x[k], y[k], x[j], y[j]);
			double d3 = d(x[i], y[i], x[j], y[j]);
			if (abs(d1 + d2 - d3) < 1e-12) n++;
		}
		int x = 1;
		rep(t, n) x = x * 2 % MOD;
		ans = (ans - x) % MOD;
	}
	ans = (ans - N - 1) % MOD;
	cout << (ans + MOD) % MOD << endl;
}

### Prompt

Your challenge is to write 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>
#include <random>
using namespace std;

#define rep(i, N) for (int i = 0; i < N; i++)
#define pb push_back
typedef long long ll;
typedef pair<int, int> i_i;
typedef pair<ll, int> ll_i;
const int MOD = 998244353;
const int INF = INT_MAX / 2;
struct edge { int v, w; };
#define double long double

double d(int x1, int y1, int x2, int y2) {
	double dx = x1 - x2, dy = y1 - y2;
	return sqrt(dx * dx + dy * dy);
}

int main() {
	int N; cin >> N;
	vector<int> x(N), y(N);
	rep(i, N) cin >> x[i] >> y[i];
	int ans = 1;
	rep(t, N) ans = ans * 2 % MOD;
	rep(i, N) rep(j, i) {
		int n = 0;
		rep(k, N) if (k != i && k != j) {
			double d1 = d(x[k], y[k], x[i], y[i]);
			double d2 = d(x[k], y[k], x[j], y[j]);
			double d3 = d(x[i], y[i], x[j], y[j]);
			if (abs(d1 + d2 - d3) < 1e-12) n++;
		}
		int x = 1;
		rep(t, n) x = x * 2 % MOD;
		ans = (ans - x) % MOD;
	}
	ans = (ans - N - 1) % MOD;
	cout << (ans + MOD) % MOD << endl;
}

```