#include <iostream>
#include <vector>
using namespace std;

struct Point { long long px, py; };

long long n, mod = 998244353; Point P[1009]; bool used[209][209];

bool solve(Point A, Point B, Point C) {
	if ((B.px - A.px)*(C.py - A.py) == (B.py - A.py)*(C.px - A.px)) return true;
	return false;
}

int main() {
	cin >> n;
	for (int i = 1; i <= n; i++) {
		cin >> P[i].px >> P[i].py;
	}
	long long ans = 1; for (int i = 1; i <= n; i++) { ans *= 2; ans %= mod; }
	ans += (mod - n - 1); ans %= mod;
	for (int i = 1; i <= n; i++) {
		for (int j = i + 1; j <= n; j++) {
			if (used[i][j] == true) continue;
			vector<int>G = { i,j };
			for (int k = j + 1; k <= n; k++) {
				if (solve(P[i], P[j], P[k]) == true) G.push_back(k);
			}
			for (int k = 0; k < G.size(); k++) {
				for (int l = k + 1; l < G.size(); l++) used[G[k]][G[l]] = true;
			}
			long long Y = 1; for (int k = 0; k < G.size(); k++) { Y *= 2; Y %= mod; }
			Y += (mod - G.size() - 1); Y %= mod;
			ans -= Y; ans += mod; ans %= mod;
		}
	}
	cout << ans << endl;
	return 0;
}
### Prompt

Construct a Cpp code solution to the problem outlined:

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 <iostream>
#include <vector>
using namespace std;

struct Point { long long px, py; };

long long n, mod = 998244353; Point P[1009]; bool used[209][209];

bool solve(Point A, Point B, Point C) {
	if ((B.px - A.px)*(C.py - A.py) == (B.py - A.py)*(C.px - A.px)) return true;
	return false;
}

int main() {
	cin >> n;
	for (int i = 1; i <= n; i++) {
		cin >> P[i].px >> P[i].py;
	}
	long long ans = 1; for (int i = 1; i <= n; i++) { ans *= 2; ans %= mod; }
	ans += (mod - n - 1); ans %= mod;
	for (int i = 1; i <= n; i++) {
		for (int j = i + 1; j <= n; j++) {
			if (used[i][j] == true) continue;
			vector<int>G = { i,j };
			for (int k = j + 1; k <= n; k++) {
				if (solve(P[i], P[j], P[k]) == true) G.push_back(k);
			}
			for (int k = 0; k < G.size(); k++) {
				for (int l = k + 1; l < G.size(); l++) used[G[k]][G[l]] = true;
			}
			long long Y = 1; for (int k = 0; k < G.size(); k++) { Y *= 2; Y %= mod; }
			Y += (mod - G.size() - 1); Y %= mod;
			ans -= Y; ans += mod; ans %= mod;
		}
	}
	cout << ans << endl;
	return 0;
}
```