#include <bits/stdc++.h>

using namespace std;

using ll = long long;

const ll MOD = 998244353;

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

int main() {
	cin.tie(0);
	ios::sync_with_stdio(false);
	int n;
	cin >> n;
	vector<int> x(n), y(n);
	for (int i = 0; i < n; i++) {
		cin >> x[i] >> y[i];
	}

	ll ans = 0;
	for (int i = 0; i < n; i++) {
		for (int j = i + 1; j < n; j++) {
			int cnt = 0;
			for (int k = j + 1; k < n; k++) {
				if (x[i] == x[j]) {
					if (x[k] == x[i]) {
						cnt++;
					}
				} else if ((x[i] - x[j]) * (y[k] - y[i]) == (y[i] - y[j]) * (x[k] - x[i])) {
					cnt++;
				}
			}
			(ans += modpow(2, cnt)) %= MOD;
		}
	}
	(ans += n + 1) % MOD;
	ans = (modpow(2, n) - ans + MOD) % MOD;
	cout << ans << endl;
	return 0;
}
### Prompt

Please provide a cpp coded solution to the problem described below:

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;

const ll MOD = 998244353;

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

int main() {
	cin.tie(0);
	ios::sync_with_stdio(false);
	int n;
	cin >> n;
	vector<int> x(n), y(n);
	for (int i = 0; i < n; i++) {
		cin >> x[i] >> y[i];
	}

	ll ans = 0;
	for (int i = 0; i < n; i++) {
		for (int j = i + 1; j < n; j++) {
			int cnt = 0;
			for (int k = j + 1; k < n; k++) {
				if (x[i] == x[j]) {
					if (x[k] == x[i]) {
						cnt++;
					}
				} else if ((x[i] - x[j]) * (y[k] - y[i]) == (y[i] - y[j]) * (x[k] - x[i])) {
					cnt++;
				}
			}
			(ans += modpow(2, cnt)) %= MOD;
		}
	}
	(ans += n + 1) % MOD;
	ans = (modpow(2, n) - ans + MOD) % MOD;
	cout << ans << endl;
	return 0;
}
```