#include <bits/stdc++.h>

const int MAXN = 210;
const int mod = 998244353;
typedef long long LL;
void reduce(int & x) { x += x >> 31 & mod; }
int mul(int a, int b) { return (LL) a * b % mod; }
int pow(int a, int b, int res = 1) {
	for (; b; b >>= 1, a = mul(a, a)) if (b & 1) res = mul(res, a);
	return res;
}
int n, xs[MAXN], ys[MAXN];
int main() {
	std::ios_base::sync_with_stdio(false), std::cin.tie(0);
	std::cin >> n;
	for (int i = 1; i <= n; ++i) std::cin >> xs[i] >> ys[i];
	int ans = pow(2, n);
	reduce(ans -= n + 1);
	for (int i = 1; i <= n; ++i)
		for (int j = i + 1; j <= n; ++j) {
			int bak = 0;
			for (int k = j + 1; k <= n; ++k)
				if ((xs[k] - xs[i]) * (ys[k] - ys[j]) == (xs[k] - xs[j]) * (ys[k] - ys[i]))
					++bak;
			reduce(ans -= pow(2, bak));
		}
	std::cout << ans << std::endl;
	return 0;
}

### Prompt

Generate 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>

const int MAXN = 210;
const int mod = 998244353;
typedef long long LL;
void reduce(int & x) { x += x >> 31 & mod; }
int mul(int a, int b) { return (LL) a * b % mod; }
int pow(int a, int b, int res = 1) {
	for (; b; b >>= 1, a = mul(a, a)) if (b & 1) res = mul(res, a);
	return res;
}
int n, xs[MAXN], ys[MAXN];
int main() {
	std::ios_base::sync_with_stdio(false), std::cin.tie(0);
	std::cin >> n;
	for (int i = 1; i <= n; ++i) std::cin >> xs[i] >> ys[i];
	int ans = pow(2, n);
	reduce(ans -= n + 1);
	for (int i = 1; i <= n; ++i)
		for (int j = i + 1; j <= n; ++j) {
			int bak = 0;
			for (int k = j + 1; k <= n; ++k)
				if ((xs[k] - xs[i]) * (ys[k] - ys[j]) == (xs[k] - xs[j]) * (ys[k] - ys[i]))
					++bak;
			reduce(ans -= pow(2, bak));
		}
	std::cout << ans << std::endl;
	return 0;
}

```