Hopping Hearts

Problem statement

N-winged rabbit is on a balance beam of length L-1. The initial position of the i-th rabbit is the integer x_i, which satisfies 0 ≤ x_ {i} \ lt x_ {i + 1} ≤ L−1. The coordinates increase as you move to the right. Any i-th rabbit can jump to the right (ie, move from x_i to x_i + a_i) any number of times, just a distance a_i. However, you cannot jump over another rabbit or enter a position below -1 or above L. Also, at most one rabbit can jump at the same time, and at most one rabbit can exist at a certain coordinate.

How many possible states of x_ {0},…, x_ {N−1} after starting from the initial state and repeating the jump any number of times? Find by the remainder divided by 1 , 000 , 000 , 007.

input

The input is given in the following format.

N L x_ {0}… x_ {N−1} a_ {0}… a_ {N−1}

Constraint

• All inputs are integers
• 1 \ ≤ N \ ≤ 5 ,000
• N \ ≤ L \ ≤ 5 ,000
• 0 \ ≤ x_ {i} \ lt x_ {i + 1} \ ≤ L−1
• 0 \ ≤ a_ {i} \ ≤ L−1

output

Print the answer in one line.

sample

Sample input 1

13 0 1

Sample output 1

3

If 1/0 is used to express the presence / absence of a rabbit, there are three ways: 100, 010, and 001.

Sample input 2

twenty four 0 1 1 2

Sample output 2

Four

There are four ways: 1100, 1001, 0101, 0011.

Sample input 3

10 50 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1

Sample output 3

272278100

The binomial coefficient C (50,10) = 10 , 272 , 278 , 170, and the remainder obtained by dividing it by 1 , 000 , 000 , 007 is 272 , 278 , 100.

Example

Input

1 3 0 1

Output

3

inputFormat

input

The input is given in the following format.

N L x_ {0}… x_ {N−1} a_ {0}… a_ {N−1}

Constraint

• All inputs are integers
• 1 \ ≤ N \ ≤ 5 ,000
• N \ ≤ L \ ≤ 5 ,000
• 0 \ ≤ x_ {i} \ lt x_ {i + 1} \ ≤ L−1
• 0 \ ≤ a_ {i} \ ≤ L−1

outputFormat

output

Print the answer in one line.

sample

Sample input 1

13 0 1

Sample output 1

3

If 1/0 is used to express the presence / absence of a rabbit, there are three ways: 100, 010, and 001.

Sample input 2

twenty four 0 1 1 2

Sample output 2

Four

There are four ways: 1100, 1001, 0101, 0011.

Sample input 3

10 50 0 1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1

Sample output 3

272278100

The binomial coefficient C (50,10) = 10 , 272 , 278 , 170, and the remainder obtained by dividing it by 1 , 000 , 000 , 007 is 272 , 278 , 100.

Example

Input

1 3 0 1

Output

3

样例

1 3
0
1

3