Given a rooted tree with the root at vertex $$$1$$$. For any vertex $$$i$$$ ($$$1 < i \leq n$$$) in the tree, there is an edge connecting vertices $$$i$$$ and $$$p_i$$$ ($$$1 \leq p_i < i$$$), with a weight equal to $$$t_i$$$.

Iris does not know the values of $$$t_i$$$, but she knows that $$$\displaystyle\sum_{i=2}^n t_i = w$$$ and each of the $$$t_i$$$ is a non-negative integer.

The vertices of the tree are numbered in a special way: the numbers of the vertices in each subtree are consecutive integers. In other words, the vertices of the tree are numbered in the order of a depth-first search.

The tree in this picture satisfies the condition. For example, in the subtree of vertex $$$2$$$, the vertex numbers are $$$2, 3, 4, 5$$$, which are consecutive integers.

The tree in this picture does not satisfy the condition, as in the subtree of vertex $$$2$$$, the vertex numbers $$$2$$$ and $$$4$$$ are not consecutive integers.

We define $$$\operatorname{dist}(u, v)$$$ as the length of the simple path between vertices $$$u$$$ and $$$v$$$ in the tree.

Next, there will be $$$n - 1$$$ events:

- Iris is given integers $$$x$$$ and $$$y$$$, indicating that $$$t_x = y$$$.

After each event, Iris wants to know the maximum possible value of $$$\operatorname{dist}(i, i \bmod n + 1)$$$ independently for each $$$i$$$ ($$$1\le i\le n$$$). She only needs to know the sum of these $$$n$$$ values. Please help Iris quickly get the answers.

Note that when calculating the maximum possible values of $$$\operatorname{dist}(i, i \bmod n + 1)$$$ and $$$\operatorname{dist}(j, j \bmod n + 1)$$$ for $$$i \ne j$$$, the unknown edge weights may be different.

Input format:
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$w$$$ ($$$2 \le n \le 2 \cdot 10^5$$$, $$$0 \leq w \leq 10^{12}$$$) — the number of vertices in the tree and the sum of the edge weights.

The second line of each test case contains $$$n - 1$$$ integers $$$p_2, p_3, \ldots, p_n$$$ ($$$1 \leq p_i < i$$$) — the description of the edges of the tree.

Then follow $$$n-1$$$ lines indicating the events. Each line contains two integers $$$x$$$ and $$$y$$$ ($$$2 \leq x \leq n$$$, $$$0 \leq y \leq w$$$), indicating that $$$t_x = y$$$.

It is guaranteed that all $$$x$$$ in the events are distinct. It is also guaranteed that the sum of all $$$y$$$ equals $$$w$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output format:
For each test case, output one line containing $$$n-1$$$ integers, each representing the answer after each event.
Example Input 0:
4
2 1000000000000
1
2 1000000000000
4 9
1 1 1
2 2
4 4
3 3
6 100
1 2 3 2 1
6 17
3 32
2 4
4 26
5 21
10 511
1 2 2 4 2 1 1 8 8
3 2
6 16
10 256
9 128
2 1
5 8
8 64
4 4
7 32

Example Output 0:
2000000000000
25 18 18
449 302 247 200 200
4585 4473 2681 1567 1454 1322 1094 1022 1022

Notes:
In the first test case, $$$\operatorname{dist}(1, 2) = \operatorname{dist}(2, 1) = t_2 = w = 10^{12}$$$, so $$$\operatorname{dist}(1, 2) + \operatorname{dist}(2, 1) = 2 \cdot 10^{12}$$$.

In the second test case, the tree after Iris found out all $$$t_x$$$ is shown below:

$$$\operatorname{dist}(1, 2) = t_2$$$, $$$\operatorname{dist}(2, 3) = t_2 + t_3$$$, $$$\operatorname{dist}(3, 4) = t_3 + t_4$$$, $$$\operatorname{dist}(4, 1) = t_4$$$. After the first event, she found out that $$$t_2 = 2$$$, so $$$\operatorname{dist}(1, 2) = 2$$$. At the same time:

- $$$\operatorname{dist}(2, 3)$$$ is maximized if $$$t_3 = 7$$$, $$$t_4 = 0$$$. Then $$$\operatorname{dist}(2, 3) = 9$$$.
- $$$\operatorname{dist}(3, 4)$$$ and $$$\operatorname{dist}(4, 1)$$$ are maximized if $$$t_3 = 0$$$, $$$t_4 = 7$$$. Then $$$\operatorname{dist}(3, 4) = \operatorname{dist}(4, 1) = 7$$$.

Thus, the answer is $$$2 + 9 + 7 + 7 = 25$$$.

After the second event, she found out that $$$t_4 = 4$$$, then $$$t_3 = w - t_2 - t_4 = 4$$$. $$$\operatorname{dist}(1, 2) = 2$$$, $$$\operatorname{dist}(2, 3) = 2 + 3 = 5$$$, $$$\operatorname{dist}(3, 4) = 3 + 4 = 7$$$, $$$\operatorname{dist}(4, 1) = 4$$$. Thus, the answer is $$$2 + 5 + 7 + 4 = 18$$$.

Example input:
4
2 1000000000000
1
2 1000000000000
4 9
1 1 1
2 2
4 4
3 3
6 100
1 2 3 2 1
6 17
3 32
2 4
4 26
5 21
10 511
1 2 2 4 2 1 1 8 8
3 2
6 16
10 256
9 128
2 1
5 8
8 64
4 4
7 32


Example output:
2000000000000
25 18 18
449 302 247 200 200
4585 4473 2681 1567 1454 1322 1094 1022 1022