Problem:
Score: 500 points

Problem Statement

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.

Sample Input 1

3
1 2 300
3 3 600
1 4 800

Sample Output 1

2900
900
0
0

When K = 0, the residents of Area 1, 2, and 3 have to walk the distances of 1, 3, and 1, respectively, to reach a railroad.

Thus, the sum of the walking distances of all citizens, S, is 1 \times 300 + 3 \times 600 + 1 \times 800 = 2900.

When K = 1, if we build a railroad along the street corresponding to the line y = 4 in the coordinate plane, the walking distances of the citizens of Area 1, 2, and 3 become 1, 1, and 0, respectively.

Then, S = 1 \times 300 + 1 \times 600 + 0 \times 800 = 900.

We have many other options for where we build the railroad, but none of them makes S less than 900.

When K = 2, if we build a railroad along the street corresponding to the lines x = 1 and x = 3 in the coordinate plane, all citizens can reach a railroad with the walking distance of 0, so S = 0. We can also have S = 0 when K = 3.

The figure below shows the optimal way to build railroads for the cases K = 0, 1, 2.

The street painted blue represents the roads along which we build railroads.

Sample Input 2

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

Sample Output 2

13800
1600
0
0
0
0

The figure below shows the optimal way to build railroads for the cases K = 1, 2.

Sample Input 3

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

Sample Output 3

26700
13900
3200
1200
0
0
0

The figure below shows the optimal way to build railroads for the case K = 3.

Sample Input 4

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

Sample Output 4

2576709
1569381
868031
605676
366338
141903
0
0
0

The figure below shows the optimal way to build railroads for the case K = 4.

Solution in C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

public class AtCoderA
{
    static void Main()
    {
        var sc = new Scanner();

        long n = sc.NextLong();
        Tuple<long, long, long>[] towns = new Tuple<long, long, long>[n];
        for (int i = 0; i < n; i++)
        {
            towns[i] = Tuple.Create(sc.NextLong(), sc.NextLong(), sc.NextLong());
        }

        towns = towns.OrderByDescending(t => t.Item3).ToArray();

        List<long> xs = new List<long>();
        List<long> ys = new List<long>();

        xs.Add(0);
        ys.Add(0);

        for (long k = 0; k <= n; k++)
        {
            long answer = long.MaxValue;

            if (k == 0)
            {
                long zeroAnswer = 0;
                for (long t = 0; t < n; t++)
                {
                    var town = towns[t];
                    long s = long.MaxValue;
                    foreach (var x in xs)
                    {
                        var tmp = town.Item3 * Math.Abs(town.Item1 - x);
                        s = Math.Min(s, tmp);
                    }
                    foreach (var y in ys)
                    {
                        var tmp = town.Item3 * Math.Abs(town.Item2 - y);
                        s = Math.Min(s, tmp);
                    }
                    zeroAnswer += s;
                }
                Console.WriteLine(zeroAnswer);
                continue;
            }

            List<long> tmpAnswerXs = new List<long>();
            List<long> tmpAnswerYs = new List<long>();
            for (int i = 0; i < n; i++)
            {
                long tmpAnswer = 0;
                List<long> tmpXs = new List<long>();
                List<long> tmpYs = new List<long>();
                for (int j = 0; j < 2; j++)
                {
                    tmpXs = new List<long>();
                    foreach (var x in xs)
                    {
                        tmpXs.Add(x);
                    }
                    tmpYs = new List<long>();
                    foreach (var y in ys)
                    {
                        tmpYs.Add(y);
                    }
                    if (j == 0)
                    {
                        tmpXs.Add(towns[i].Item1);
                    }
                    else
                    {
                        tmpYs.Add(towns[i].Item2);
                    }

                    for (long t = 0; t < n; t++)
                    {
                        var town = towns[t];
                        long s = long.MaxValue;
                        foreach (var x in tmpXs)
                        {
                            var tmp = town.Item3 * Math.Abs(town.Item1 - x);
                            s = Math.Min(s, tmp);
                        }
                        foreach (var y in tmpYs)
                        {
                            var tmp = town.Item3 * Math.Abs(town.Item2 - y);
                            s = Math.Min(s, tmp);
                        }
                        tmpAnswer += s;
                    }
                    if (tmpAnswer < answer)
                    {
                        answer = tmpAnswer;
                        tmpAnswerXs = new List<long>();
                        foreach (var x in tmpXs)
                        {
                            tmpAnswerXs.Add(x);
                        }
                        tmpAnswerYs = new List<long>();
                        foreach (var y in tmpYs)
                        {
                            tmpAnswerYs.Add(y);
                        }
                    }
                }
            }

            Console.WriteLine(answer);
            foreach (var x in tmpAnswerXs)
            {
                if (!xs.Contains(x))
                {
                    xs.Add(x);
                }
            }
            foreach (var y in tmpAnswerYs)
            {
                if (!ys.Contains(y))
                {
                    ys.Add(y);
                }
            }
        }
    }

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            while (_index >= _line.Length)
            {
                _line = Console.ReadLine().Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            _line = Console.ReadLine().Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}