1D Matching

There are N computers and N sockets in a one-dimensional world. The coordinate of the i-th computer is a_i, and the coordinate of the i-th socket is b_i. It is guaranteed that these 2N coordinates are pairwise distinct.

Snuke wants to connect each computer to a socket using a cable. Each socket can be connected to only one computer.

In how many ways can he minimize the total length of the cables? Compute the answer modulo 10^9+7.

Constraints

• 1 ≤ N ≤ 10^5
• 0 ≤ a_i, b_i ≤ 10^9
• The coordinates are integers.
• The coordinates are pairwise distinct.

Input

The input is given from Standard Input in the following format:

N a_1 : a_N b_1 : b_N

Output

Print the number of ways to minimize the total length of the cables, modulo 10^9+7.

Examples

Input

2 0 10 20 30

Output

2

Input

3 3 10 8 7 12 5

Output

1

inputFormat

Input

The input is given from Standard Input in the following format:

N a_1 : a_N b_1 : b_N

outputFormat

Output

Print the number of ways to minimize the total length of the cables, modulo 10^9+7.

Examples

Input

2 0 10 20 30

Output

2

Input

3 3 10 8 7 12 5

Output

1

样例

2
0
10
20
30

2