A. Fancy Fence
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

Everybody knows that Balázs has the fanciest fence in the whole town. It's built up from $$$N$$$ fancy sections. The sections are rectangles standing closely next to each other on the ground. The $$$i$$$th section has integer height $$$h_i$$$ and integer width $$$w_i$$$. We are looking for fancy rectangles on this fancy fence. A rectangle is fancy if:

  • its sides are either horizontal or vertical and have integer lengths
  • the distance between the rectangle and the ground is integer
  • the distance between the rectangle and the left side of the first section is integer
  • it's lying completely on sections
What is the number of fancy rectangles? This number can be very big, so we are interested in it modulo $$$10^9+7$$$.
Input

The first line contains $$$N$$$ ($$$1\leq N \leq 10^{5}$$$) – the number of sections. The second line contains $$$N$$$ space-separated integers, the $$$i$$$th number is $$$h_i$$$ ($$$1 \leq h_i \leq 10^{9}$$$). The third line contains $$$N$$$ space-separated integers, the $$$i$$$th number is $$$w_i$$$ ($$$1 \leq w_i \leq 10^{9}$$$).

Output

You should print a single integer, the number of fancy rectangles modulo $$$10^9+7$$$. So the output range is $$$0,1,2,\ldots, 10^9+6$$$.

Scoring
$$$ \begin{array}{|c|c|c|} \hline \text{Subtask} & \text{Points} & \text{Constraints} \\ \hline 1 & 0 & \text{sample}\\ \hline 2 & 12 & N \leq 50 \: \text{and} \: h_i \leq 50 \: \text{and} \: w_i = 1 \: \text{for all} \: i \\ \hline 3 & 13 & h_i = 1 \: \text{or} \: h_i = 2 \: \text{for all} \: i \\ \hline 4 & 15 & \text{all} \: h_i \: \text{are equal} \\ \hline 5 & 15 & h_i \leq h_{i+1} \: \text{for all} \: i \leq N-1 \\ \hline 6 & 18 & N \leq 1000\\ \hline 7 & 27 & \text{no additional constraints}\\ \hline \end{array} $$$
Example
Input
2
1 2
1 2
Output
12
Note

The fence looks like this:

There are 5 fancy rectangles of shape:

There are 3 fancy rectangles of shape:

There is 1 fancy rectangle of shape:

There are 2 fancy rectangles of shape:

There is 1 fancy rectangle of shape: