Problem: New AtCoder City has an infinite grid of streets, as follows:

* At the center of the city stands a clock tower. Let (0, 0) be the coordinates of this point.
* A straight street, which we will call East-West Main Street, runs east-west and passes the clock tower. It corresponds to the x-axis in the two-dimensional coordinate plane.
* There are also other infinitely many streets parallel to East-West Main Street, with a distance of 1 between them. They correspond to the lines \ldots, y = -2, y = -1, y = 1, y = 2, \ldots in the two-dimensional coordinate plane.
* A straight street, which we will call North-South Main Street, runs north-south and passes the clock tower. It corresponds to the y-axis in the two-dimensional coordinate plane.
* There are also other infinitely many streets parallel to North-South Main Street, with a distance of 1 between them. They correspond to the lines \ldots, x = -2, x = -1, x = 1, x = 2, \ldots in the two-dimensional coordinate plane.



There are N residential areas in New AtCoder City. The i-th area is located at the intersection with the coordinates (X_i, Y_i) and has a population of P_i. Each citizen in the city lives in one of these areas.

The city currently has only two railroads, stretching infinitely, one along East-West Main Street and the other along North-South Main Street.
M-kun, the mayor, thinks that they are not enough for the commuters, so he decides to choose K streets and build a railroad stretching infinitely along each of those streets.

Let the walking distance of each citizen be the distance from his/her residential area to the nearest railroad.
M-kun wants to build railroads so that the sum of the walking distances of all citizens, S, is minimized.

For each K = 0, 1, 2, \dots, N, what is the minimum possible value of S after building railroads?

Constraints

* 1 \leq N \leq 15
* -10 \ 000 \leq X_i \leq 10 \ 000
* -10 \ 000 \leq Y_i \leq 10 \ 000
* 1 \leq P_i \leq 1 \ 000 \ 000
* The locations of the N residential areas, (X_i, Y_i), are all distinct.
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N
X_1 Y_1 P_1
X_2 Y_2 P_2
:  :  :
X_N Y_N P_N


Output

Print the answer in N+1 lines.
The i-th line (i = 1, \ldots, N+1) should contain the minimum possible value of S after building railroads for the case K = i-1.

Examples

Input

3
1 2 300
3 3 600
1 4 800


Output

2900
900
0
0


Input

5
3 5 400
5 3 700
5 5 1000
5 7 700
7 5 400


Output

13800
1600
0
0
0
0


Input

6
2 5 1000
5 2 1100
5 5 1700
-2 -5 900
-5 -2 600
-5 -5 2200


Output

26700
13900
3200
1200
0
0
0


Input

8
2 2 286017
3 1 262355
2 -2 213815
1 -3 224435
-2 -2 136860
-3 -1 239338
-2 2 217647
-1 3 141903


Output

2576709
1569381
868031
605676
366338
141903
0
0
0

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
using namespace std;
#define rep(i,n) for(int i=0; i<(n); i++)
using ll = long long;
int n, x[20], y[20], p[20];
bool used[20];
ll ans[20];
set<int> xs, ys;
template<class T> void chmin(T &a, T b){if(a>b) a=b;}
void rec(int i){
	if(i == n){
		int cnt = xs.size() + ys.size() - 2;
		ll sm = 0;
		rep(i,n) if(!used[i]){
			int d = 1e9;
			auto it = xs.lower_bound(x[i]);
			if(it != xs.end()) chmin(d,(*it)-x[i]);
			if(it != xs.begin()) chmin(d,x[i]-*--it);
			it = ys.lower_bound(y[i]);
			if(it != ys.end()) chmin(d,(*it)-y[i]);
			if(it != ys.begin()) chmin(d,y[i]-*--it);
			sm += (ll)d * p[i];
		}
		if(ans[cnt] > sm) ans[cnt] = sm;
	}else{
		if(xs.count(x[i]) || ys.count(y[i])){
			used[i] = true;
			rec(i+1);
			used[i] = false;
		}else{
			rec(i+1);
			used[i] = true;
			xs.insert(x[i]);
			rec(i+1);
			xs.erase(x[i]);
			ys.insert(y[i]);
			rec(i+1);
			ys.erase(y[i]);
			used[i] = false;
		}
	}
}
int main(){
	scanf("%d", &n);
	fill(ans,ans+n+1,(ll)1e15);
	rep(i,n) scanf("%d%d%d", x+i, y+i, p+i);
	xs.insert(0); ys.insert(0);
	rec(0);
	rep(i,n) chmin(ans[i+1], ans[i]);
	rep(i,n+1) printf("%lld\n", ans[i]);
}