#include<bits/stdc++.h>
#define rep(i, l, r) for(int i = (l), i##end = (r);i <= i##end;++i)
using std::cin; using std::cout;
const int maxn = 200200;
typedef long long ll;
const int mod = 998244353;
inline ll pow(ll a,int b,int ans = 1) {
	for(;b;b >>= 1,a = a * a % mod) if(b & 1)
		ans = ans * a % mod;
	return ans;
}
inline ll inverse(int x){ return pow(x, mod - 2); }
int n;
int x[maxn], y[maxn];
int pw[maxn];
int main() {
	std::ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
	*pw = 1;
	cin >> n;
	rep(i, 1, n) {
		cin >> x[i] >> y[i];
		pw[i] = (pw[i - 1] << 1) % mod;
	}
	int ans = pw[n] - n - 1;
	ans += ans >> 31 & mod;
	rep(i, 1, n) {
		rep(j, i + 1, n) {
			int cnt = 0;
			rep(k, j + 1, n) {
				cnt += (ll) (x[j] - x[i]) * (y[k] - y[i]) == (ll) (x[k] - x[i]) * (y[j] - y[i]);
			}
			ans -= pw[cnt];
			ans += ans >> 31 & mod;
		}
	}
	cout << ans << '\n';
}

### 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>
#define rep(i, l, r) for(int i = (l), i##end = (r);i <= i##end;++i)
using std::cin; using std::cout;
const int maxn = 200200;
typedef long long ll;
const int mod = 998244353;
inline ll pow(ll a,int b,int ans = 1) {
	for(;b;b >>= 1,a = a * a % mod) if(b & 1)
		ans = ans * a % mod;
	return ans;
}
inline ll inverse(int x){ return pow(x, mod - 2); }
int n;
int x[maxn], y[maxn];
int pw[maxn];
int main() {
	std::ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
	*pw = 1;
	cin >> n;
	rep(i, 1, n) {
		cin >> x[i] >> y[i];
		pw[i] = (pw[i - 1] << 1) % mod;
	}
	int ans = pw[n] - n - 1;
	ans += ans >> 31 & mod;
	rep(i, 1, n) {
		rep(j, i + 1, n) {
			int cnt = 0;
			rep(k, j + 1, n) {
				cnt += (ll) (x[j] - x[i]) * (y[k] - y[i]) == (ll) (x[k] - x[i]) * (y[j] - y[i]);
			}
			ans -= pw[cnt];
			ans += ans >> 31 & mod;
		}
	}
	cout << ans << '\n';
}

```