Byteland coins

Description:

Input:

The first line contains single integer n (1 ≤ n ≤ 3·10⁵) — the number of coin types.

The second line contains n - 1 integers a₁, a₂, ..., a_n - 1 (1 ≤ a_k ≤ 10⁹) — the ratios between the coin types denominations. It is guaranteed that for each x there are at most 20 coin types of denomination x.

The third line contains n non-negative integers b₁, b₂, ..., b_n — the number of coins of each type Byteasar has. It is guaranteed that the sum of these integers doesn't exceed 3·10⁵.

The fourth line contains single integer m (0 ≤ m < 10¹⁰⁰⁰⁰) — the amount in tugricks Byteasar needs to pay.

Output:

Print single integer — the number of ways to pay exactly m tugricks modulo 10⁹ + 7.

Sample Input:

1

4
2

Sample Output:

1

Sample Input:

2
1
4 4
2

Sample Output:

3

Sample Input:

3
3 3
10 10 10
17

Sample Output:

6

Note:

In the first example Byteasar has 4 coins of denomination 1, and he has to pay 2 tugricks. There is only one way.

In the second example Byteasar has 4 coins of each of two different types of denomination 1, he has to pay 2 tugricks. There are 3 ways: pay one coin of the first type and one coin of the other, pay two coins of the first type, and pay two coins of the second type.

In the third example the denominations are equal to 1, 3, 9.