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 PYTHON3):

Solution:
from bisect import bisect_left

n = int(input())
xyp = [list(map(int, input().split())) for i in range(n)]
inf = 1<<63
tate = [[] for i in range(1<<n)]
yoko = [[] for i in range(1<<n)]
state_ind = [[] for i in range(1<<n)]
for state in range(1<<n):
    tmp_x = [0]
    tmp_y = [0]
    for i in range(n):
        if state & (1<<i):
            tmp_x.append(xyp[i][0])
            tmp_y.append(xyp[i][1])
        else:
            state_ind[state].append(i)
    tmp_x.sort()
    tmp_y.sort()
    x_score = [0]*n
    y_score = [0]*n
    nn = len(tmp_x)
    mm = len(tmp_y)
    for i, (x,y,p) in enumerate(xyp):
        j = bisect_left(tmp_x,x)
        if j == 0:
            x_m = tmp_x[0]-x
        elif j == nn:
            x_m = x-tmp_x[nn-1]
        else:
            x_m = min(x-tmp_x[j-1], tmp_x[j]-x)
        j = bisect_left(tmp_y,y)
        if j == 0:
            y_m = tmp_y[0]-y
        elif j == mm:
            y_m = y-tmp_y[mm-1]
        else:
            y_m = min(y-tmp_y[j-1], tmp_y[j]-y)
        x_score[i] = x_m * p
        y_score[i] = y_m * p
    tate[state] = x_score
    yoko[state] = y_score

ans = [inf]*(1<<n)
for state in range(1<<n):
    j = state
    while j>=0:
        j &= state
        i = state^j
        tmp1 = 0
        for k in state_ind[state]:
            if tate[i][k] < yoko[j][k]:
                tmp1 += tate[i][k]
            else:
                tmp1 += yoko[j][k]
            if tmp1 >= ans[state]:
                break
        if tmp1 < ans[state]:
            ans[state] = tmp1 
        j -= 1

cnt = [0]*(1<<n)
for i in range(1<<(n-1)):
    cnt[2*i] = cnt[i]
    cnt[2*i+1] = cnt[i]+1

ans_k = [inf]*(n+1)
for state in range(1<<n):
    k = cnt[state]
    if ans_k[k] > ans[state]:
        ans_k[k] = ans[state]
print(*ans_k, sep="\n")