Christmas Balls

Christmas is coming, and Alessandro has to decorate his living room with a Christmas tree.

He heard that in Pordenone a new tree shop has just opened, so he decided to go there. This shop has a huge tree decorated with $N$ balls, one at each branching point (the root is node $0$). Of course the balls are quite spectacular, and he noticed that the balls come in $C$ different colors in total.

The peculiarity of this shop is that you can buy the whole tree, or cut a branch and buy just a subtree! You can make the cut just below a ball, so that you take that ball and the subtree rooted there.

Alessandro doesn't want just a random tree, he wants the nicest. His taste is in fact quite peculiar, he'd like that the amount of balls of each color to be well distributed. Precisely, let's call $x$ the number of balls of (one of) the most frequent color in a subtree, he wants to maximize the number of colors with $x$ balls each.

Help Alessandro finding where to cut the tree by printing the maximum amount of most-frequent colors he can get.

Input data

The first line contains the only integer $N$, the number of nodes of the tree.

The second line contains $N$ integers $C_i$, the colors of each ball.

The third line contains $N-1$ integers: $P_i$ is the index of the parent of the $i$-th node.

Output data

The output contains a single line with an integer: chosen the optimal subtree, the number of colors with the highest frequency.

Constraints and Clarifications

• $1 \leq N \leq 10^5$;
• $0 \leq C_i < N$;
• For tests worth $13$ points, the tree is a line.
• For tests worth $15$ more points, $1 \leq N \leq 1 \ 000$, $1 \leq MAXC \leq 2$.
• For tests worth $21$ more points, $1 \leq N \leq 1 \ 000$.
• For tests worth $17$ more points, $1 \leq MAXC \leq 2$.

Example 1

stdin

8
1 2 0 2 0 0 1 1
0 0 1 1 3 4 4

stdout

3

Explanation

In the first sample case, there are $8$ nodes in $3$ different colors. The solution is to cut the tree and take the subtree rooted at node $1$. This way we are left with $6$ nodes: the most frequent color has $2$ balls, and there are $3$ colors with that many balls.

Example 2

stdin

5
0 1 1 0 0
0 1 2 3

stdout

2

Explanation

If, for example, we have kept the entire tree, the most frequent color appears with $3$ balls, but there are only $2$ colors with $3$ balls, so this is worse than the aforementioned solution.