#include <bits/stdc++.h>
using namespace std;
using ll = long long;

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);

    int N;
    cin >> N;
    vector<ll> X(N), Y(N), P(N);
    for (int i = 0; i < N; i++) {
        cin >> X[i] >> Y[i] >> P[i];
    }
    vector<vector<ll>> ver(1 << N, vector<ll>(N)), hor(1 << N, vector<ll>(N));
    for (int S = 0; S < 1 << N; S++) {
        for (int i = 0; i < N; i++) {
            ll v = abs(X[i]), h = abs(Y[i]);
            for (int j = 0; j < N; j++) {
                if (S >> j & 1) {
                    v = min(v, abs(X[i] - X[j]));
                    h = min(h, abs(Y[i] - Y[j]));
                }
            }
            ver[S][i] = v * P[i];
            hor[S][i] = h * P[i];
        }
    }
    vector<ll> ans(N+1, 1e18);
    for (int S = 0; S < 1 << N; S++) {
        int cnt = 0;
        int v = 0, h = 0;
        vector<int> vv, hh;
        for (int i = 0; i < N; i++) {
            if (S >> i & 1) {
                cnt++;
                vv.push_back(1 << i);
                hh.push_back(1 << i);
            }
        }
        for (int b = 0; b < 1 << cnt; b++) {
            int v = 0, h = 0;
            for (int i = 0; i < cnt; i++) {
                if (b >> i & 1) v += vv[i];
                else h += hh[i];
            }

            ll cost = 0;
            for (int i = 0; i < N; i++) {
                cost += min(ver[v][i], hor[h][i]);
            }
            ans[cnt] = min(ans[cnt], cost);
        }
    }
    for (int i = 0; i <= N; i++) cout << ans[i] << '\n';
}
### Prompt

Please formulate a cpp solution to the following 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

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
using ll = long long;

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);

    int N;
    cin >> N;
    vector<ll> X(N), Y(N), P(N);
    for (int i = 0; i < N; i++) {
        cin >> X[i] >> Y[i] >> P[i];
    }
    vector<vector<ll>> ver(1 << N, vector<ll>(N)), hor(1 << N, vector<ll>(N));
    for (int S = 0; S < 1 << N; S++) {
        for (int i = 0; i < N; i++) {
            ll v = abs(X[i]), h = abs(Y[i]);
            for (int j = 0; j < N; j++) {
                if (S >> j & 1) {
                    v = min(v, abs(X[i] - X[j]));
                    h = min(h, abs(Y[i] - Y[j]));
                }
            }
            ver[S][i] = v * P[i];
            hor[S][i] = h * P[i];
        }
    }
    vector<ll> ans(N+1, 1e18);
    for (int S = 0; S < 1 << N; S++) {
        int cnt = 0;
        int v = 0, h = 0;
        vector<int> vv, hh;
        for (int i = 0; i < N; i++) {
            if (S >> i & 1) {
                cnt++;
                vv.push_back(1 << i);
                hh.push_back(1 << i);
            }
        }
        for (int b = 0; b < 1 << cnt; b++) {
            int v = 0, h = 0;
            for (int i = 0; i < cnt; i++) {
                if (b >> i & 1) v += vv[i];
                else h += hh[i];
            }

            ll cost = 0;
            for (int i = 0; i < N; i++) {
                cost += min(ver[v][i], hor[h][i]);
            }
            ans[cnt] = min(ans[cnt], cost);
        }
    }
    for (int i = 0; i <= N; i++) cout << ans[i] << '\n';
}
```