Split the Tree

You are given a rooted tree on n vertices, its root is the vertex number 1. The i-th vertex contains a number w_i. Split it into the minimum possible number of vertical paths in such a way that each path contains no more than L vertices and the sum of integers w_i on each path does not exceed S. Each vertex should belong to exactly one path.

A vertical path is a sequence of vertices v_1, v_2, …, v_k where v_i (i ≥ 2) is the parent of v_{i - 1}.

Input

The first line contains three integers n, L, S (1 ≤ n ≤ 10^5, 1 ≤ L ≤ 10^5, 1 ≤ S ≤ 10^{18}) — the number of vertices, the maximum number of vertices in one path and the maximum sum in one path.

The second line contains n integers w_1, w_2, …, w_n (1 ≤ w_i ≤ 10^9) — the numbers in the vertices of the tree.

The third line contains n - 1 integers p_2, …, p_n (1 ≤ p_i < i), where p_i is the parent of the i-th vertex in the tree.

Output

Output one number — the minimum number of vertical paths. If it is impossible to split the tree, output -1.

Examples

Input

3 1 3 1 2 3 1 1

Output

3

Input

3 3 6 1 2 3 1 1

Output

2

Input

1 1 10000 10001

Output

-1

Note

In the first sample the tree is split into {1},\ {2},\ {3}.

In the second sample the tree is split into {1,\ 2},\ {3} or {1,\ 3},\ {2}.

In the third sample it is impossible to split the tree.

inputFormat

Input

The first line contains three integers n, L, S (1 ≤ n ≤ 10^5, 1 ≤ L ≤ 10^5, 1 ≤ S ≤ 10^{18}) — the number of vertices, the maximum number of vertices in one path and the maximum sum in one path.

The second line contains n integers w_1, w_2, …, w_n (1 ≤ w_i ≤ 10^9) — the numbers in the vertices of the tree.

The third line contains n - 1 integers p_2, …, p_n (1 ≤ p_i < i), where p_i is the parent of the i-th vertex in the tree.

outputFormat

Output

Output one number — the minimum number of vertical paths. If it is impossible to split the tree, output -1.

Examples

Input

3 1 3 1 2 3 1 1

Output

3

Input

3 3 6 1 2 3 1 1

Output

2

Input

1 1 10000 10001

Output

-1

Note

In the first sample the tree is split into {1},\ {2},\ {3}.

In the second sample the tree is split into {1,\ 2},\ {3} or {1,\ 3},\ {2}.

In the third sample it is impossible to split the tree.

样例

3 1 3
1 2 3
1 1

3

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