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문제

Bobo has a rooted tree with $n$ nodes which are conveniently labeled with $1, 2, \dots, n$. Node $1$ is the root, and the $i$-th node has weight $w_i$.

He would like to find out $f(2), f(3), \dots, f(n)$ where

$$f(i) = \sum_{j = 1}^{i - 1} w_{\mathrm{LCA}(i, j)}\text{.}$$

입력

The input contains zero or more test cases, and is terminated by end-of-file. For each test case:

The first line contains an integer $n$ ($2 \leq n \leq 2 \cdot 10^5$).

The second line contains $n$ integers $w_1, w_2, \dots, w_n$ ($1 \leq w_i \leq 10^4$).

The third line contains $(n - 1)$ integers $p_2, p_3, \dots, p_n$, where $p_i$ denotes an edge from the $p_i$-th node to the $i$-th node ($1 \leq p_i \leq n$). The edges form a tree.

It is guaranteed that the sum of $n$ does not exceed $2 \cdot 10^5$.

출력

For each test case, output $(n - 1)$ integers: $f(2), f(3), \dots, f(n)$.

예제 입력 1

3
1 2 3
1 1
5
1 2 3 4 5
1 2 2 1

예제 출력 1

1
2
1
3
5
4