Problem Statement

There are $N$ continuous random variables $x_i$ ($1 ≤ i ≤ N$), each of which follows the continuous uniform distribution on the interval \[L_i, R_i\]. (That is, $x_i$ is a random variable that can take any real value between $L_i$ and $R_i$ (inclusive) with equal probability.)

Let $E$ be the expected value of the maximum of these $N$ random variables. Under the constraints in this problem, it can be proved that $E \\times (N+1)! \\times \\prod_{i=1}^N (R_i - L_i)$ is a positive integer. Find this value modulo $1,000,000,007$.

Constraints

• $1 ≤ N ≤ 1000$
• $0 ≤ L_i < R_i ≤ 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $L_1$ $R_1$ $:$ $L_N$ $R_N$

Output

Print $E \\times (N+1)! \\times \\prod_{i=1}^N (R_i - L_i)$ modulo $1,000,000,007$, as an integer.

Sample Input 1

1
1 2

Sample Output 1

3

The expected value of the maximum of these random variables - there is actually just one in this case - is equal to the median of the range of the values the variable can take, that is, $E = \\frac{3}{2}$.

Thus, the correct output is $E \\times (N+1)! \\times (R_1 - L_1) = E \\times 2 = 3$.

Sample Input 2

2
1 2
1 2

Sample Output 2

10

The expected value in question is $E = \\frac{5}{3}$.

Sample Input 3

2
1 2
2 4

Sample Output 3

36

Sample Input 4

5
40 96
81 92
16 384
32 768
65 536

Sample Output 4

52776507