F - Expensive Expense /

### 問題文

AtCoder 王国は N 個の街と N-1 個の道路からなります。

• 厳密に言い換えると、i - j 間の最短パスを i = p_0, p_1, \dots, p_{k-1}, p_k = j として、街 p_{l} と街 p_{l+1} を結ぶ道路の通行料を c_l と置いたときに E_{i,j} = D_j + \displaystyle\sum_{l=0}^{k-1} c_l と定義します。

すべての i に対して、街 i を始点として他の街へ旅行したときにありえる旅費の最大値を求めてください。

• 厳密に言い換えると、すべての i に対して \max_{1 \leq j \leq N, j \neq i} E_{i,j} を求めてください。

### 制約

• 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)
• 整数の組 (i,j)1 \leq i \lt j \leq N を満たすならば、街 i から街 j へいくつかの道路を通ることで移動できる。
• 入力はすべて整数である。

### 入力

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


### 出力

N 行出力せよ。i 行目には \displaystyle \max_{1 \leq j \leq N, j \neq i} E_{i,j} を出力せよ。

### 入力例 1

3
1 2 2
2 3 3
1 2 3


### 出力例 1

8
6
6


すべての街の順序つき組 (i,j) に対して E_{i,j} を計算すると次のようになります。

• 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

### 入力例 2

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


### 出力例 2

105
108
106
109
106
14


### 入力例 3

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


### 出力例 3

5000000006
4000000006
3000000006
3000000001
4000000001
5000000001


Score : 500 points

### 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}.

### 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.

### 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}.

### 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