You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in c++17.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```cpp
<your code here>
```

Execution time limit: 7.0 seconds
Memory limit: 256.0 MB

# Problem
Mr. Chanek has an orchard structured as a rooted ternary tree with $$$N$$$ vertices numbered from $$$1$$$ to $$$N$$$. The root of the tree is vertex $$$1$$$. $$$P_i$$$ denotes the parent of vertex $$$i$$$, for $$$(2 \le i \le N)$$$. Interestingly, the height of the tree is not greater than $$$10$$$. Height of a tree is defined to be the largest distance from the root to a vertex in the tree.

There exist a bush on each vertex of the tree. Initially, all bushes have fruits. Fruits will not grow on bushes that currently already have fruits. The bush at vertex $$$i$$$ will grow fruits after $$$A_i$$$ days since its last harvest.

Mr. Chanek will visit his orchard for $$$Q$$$ days. In day $$$i$$$, he will harvest all bushes that have fruits on the subtree of vertex $$$X_i$$$. For each day, determine the sum of distances from every harvested bush to $$$X_i$$$, and the number of harvested bush that day. Harvesting a bush means collecting all fruits on the bush.

For example, if Mr. Chanek harvests all fruits on subtree of vertex $$$X$$$, and harvested bushes $$$[Y_1, Y_2, \dots, Y_M]$$$, the sum of distances is $$$\sum_{i = 1}^M \text{distance}(X, Y_i)$$$

$$$\text{distance}(U, V)$$$ in a tree is defined to be the number of edges on the simple path from $$$U$$$ to $$$V$$$.

## Input Format
The first line contains two integers $$$N$$$ and $$$Q$$$ $$$(1 \le N,\ Q,\le 5 \cdot 10^4)$$$, which denotes the number of vertices and the number of days Mr. Chanek visits the orchard.

The second line contains $$$N$$$ integers $$$A_i$$$ $$$(1 \le A_i \le 5 \cdot 10^4)$$$, which denotes the fruits growth speed on the bush at vertex $$$i$$$, for $$$(1 \le i \le N)$$$.

The third line contains $$$N-1$$$ integers $$$P_i$$$ $$$(1 \le P_i \le N, P_i \ne i)$$$, which denotes the parent of vertex $$$i$$$ in the tree, for $$$(2 \le i \le N)$$$. It is guaranteed that each vertex can be the parent of at most $$$3$$$ other vertices. It is also guaranteed that the height of the tree is not greater than $$$10$$$.

The next $$$Q$$$ lines contain a single integer $$$X_i$$$ $$$(1 \le X_i \le N)$$$, which denotes the start of Mr. Chanek's visit on day $$$i$$$, for $$$(1 \le i \le Q)$$$.

## Output Format
Output $$$Q$$$ lines, line $$$i$$$ gives the sum of distances from the harvested bushes to $$$X_i$$$, and the number of harvested bushes.

## Examples
```input
2 3
1 2
1
2
1
1
```
```output
0 1
0 1
1 2
```
-----
```input
5 3
2 1 1 3 2
1 2 2 1
1
1
1
```
```output
6 5
3 2
4 4
```

## Note
For the first example:

- On day 1, Mr. Chanek starts at vertex $$$2$$$ and can harvest the bush at vertex 2.
- On day 2, Mr. Chanek starts at vertex $$$1$$$ and only harvest from bush $$$1$$$ (bush 2's fruit still has not grown yet).
- On day 3, Mr. Chanek starts at vertex $$$1$$$ and harvests the fruits on bush $$$1$$$ and $$$2$$$. The sum of distances from every harvested bush to $$$1$$$ is $$$1$$$.

For the second example, Mr. Chanek always starts at vertex $$$1$$$. The bushes which Mr. Chanek harvests on day one, two, and three are $$$[1, 2, 3, 4, 5], [2, 3], [1, 2, 3, 5]$$$, respectively.

Now solve the problem and return the code.