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 PHP (7.4.4) (judge verdict: Time Limit Exceeded)

<?php
list($n) = ints();
for ($i = 0; $i < $n; ++$i) {
    list($x[$i], $y[$i], $p[$i]) = ints();
    /*    if (isset($cntx[$x[$i]])) ++$cntx[$x[$i]];
    else $cntx[$x[$i]] = 1;
    if (isset($cnty[$y[$i]])) ++$cnty[$y[$i]];
    else $cnty[$y[$i]] = 1;*/
}
for ($k = 0; $k <= $n; ++$k) $ans[] = dfs($k);
echo implode(PHP_EOL, $ans);
function dfs($k = 0, $d = 0, $lxs = [true], $lys = [true])
{
    global $n, $x, $y, $cntx, $cnty;
    if (count($lxs) + count($lys) - 2 > $k) return PHP_INT_MAX;
    if ($d === $n || count($lxs) + count($lys) - 2 === $k) return calc($lxs, $lys);
    $res[] = dfs($k, $d + 1, $lxs, $lys);
    $_lxs = $lxs;
    $_lys = $lys;
    $_lxs[$x[$d]] = true;
    $_lys[$y[$d]] = true;
    $res[] = dfs($k, $d + 1, $_lxs, $lys);
    $res[] = dfs($k, $d + 1, $lxs, $_lys);
    $res[] = dfs($k, $d + 1, $_lxs, $_lys);
    return min($res);
}
function calc($lxs, $lys)
{
    global $n, $x, $y, $p;
    $res = 0;
    $lxs = array_keys($lxs);
    $lys = array_keys($lys);
    for ($i = 0; $i < $n; ++$i) {
        $dist = PHP_INT_MAX;
        foreach ($lxs as $lx)
            $dist = min($dist, abs($x[$i] - $lx));
        foreach ($lys as $ly)
            $dist = min($dist, abs($y[$i] - $ly));
        $res += $dist * $p[$i];
    }
    return $res;
}
function ints()
{
    return array_map('intval', explode(' ', trim(fgets(STDIN))));
}