Problem Statement

The Kingdom of AtCoder is composed of N towns and N-1 roads.
The towns are labeled as Town $1, Town 2, \dots, Town N$.
Similarly, the roads are labeled as Road $1, Road 2, \dots, Road N-1$.
Road i connects Towns A_i and B_i bidirectionally, and you have to pay the toll of C_i to go through it. For every pair of different towns (i, j), it is possible to go from Town A_i to Town B_j via the roads.
You are given $a sequence D = (D_1, D_2, \dots, D_N)$, where D_i is the cost to do sightseeing in Town i.
Let us define the travel cost E_{i,j} from Town i to Town j as the total toll incurred when going from Town i to Town j, plus D_{j}.

• More formally, E_{i,j} is defined as $D_j + \displaystyle\sum_{l=0}^{k-1} c_l$, where the shortest path between i and j is $i = p_0, p_1, \dots, p_{k-1}, p_k = j$ and the toll for the road connecting Towns p_{l} and p_{l+1} is c_l.

For every i, find the maximum possible travel cost when traveling from Town i to another town.

• More formally, for every $i, find \max_{1 \leq j \leq N, j \neq i} E_{i,j}$.

Input

Input is given from Standard Input in the following format:
N
A_1 B_1 C_1
A_2 B_2 C_2
$\vdots$
A_{N-1} B_{N-1} C_{N-1}
$D_1 D_2 \dots D_N$

Output

Print N lines. The i-th line should contain $\displaystyle \max_{1 \leq j \leq N, j \neq i} E_{i,j}$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq A_i \leq N (1 \leq i \leq N-1)$
• $1 \leq B_i \leq N (1 \leq i \leq N-1)$
• $1 \leq C_i \leq 10^9 (1 \leq i \leq N-1)$
• $1 \leq D_i \leq 10^9 (1 \leq i \leq N)$
• It is possible to travel from Town i to Town j via some number of roads, for a pair of integers (i,j) such that $1 \leq i \lt j \leq N$.
• All values in input are integers.

Sample Input 1

3
1 2 2
2 3 3
1 2 3

Sample Output 1

8
6
6

The value of E_{i,j} for every ordered pair of towns (i,j) is as follows.

• E_{1,2} = 2 + 2 = 4
• E_{1,3} = 5 + 3 = 8
• E_{2,1} = 2 + 1 = 3
• E_{2,3} = 3 + 3 = 6
• E_{3,1} = 5 + 1 = 6
• E_{3,2} = 3 + 2 = 5

Sample Input 2

6
1 2 3
1 3 1
1 4 4
1 5 1
1 6 5
9 2 6 5 3 100

Sample Output 2

105
108
106
109
106
14

Sample Input 3

6
1 2 1000000000
2 3 1000000000
3 4 1000000000
4 5 1000000000
5 6 1000000000
1 2 3 4 5 6

Sample Output 3

5000000006
4000000006
3000000006
3000000001
4000000001
5000000001

见题面 Input 部分

见题面 Output 部分