SOLUTIONS Programming Contest 2020 - M's Solution

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

inputFormat

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

outputFormat

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

样例

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

2576709
1569381
868031
605676
366338
141903
0
0
0

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