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 \(10^9+7\).
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 \(w_i\) (\(1 \leq w_i \leq 10^{9}\)).
You should print a single integer, the number of fancy rectangles modulo \(10^9+7\). So the output range is \(0,1,2,\ldots, 10^9+6\).
\(\begin{array}{|c|c|c|} \hline \text{Subtask} & \text{Points} & \text{Constraints} \\ \hline 1 & 0 & \text{sample}\\ \hline 2 & 12 & N \leq 50 \: \text{and} \: h_i \leq 50 \: \text{and} \: w_i = 1 \: \text{for all} \: i \\ \hline 3 & 13 & h_i = 1 \: \text{or} \: h_i = 2 \: \text{for all} \: i \\ \hline 4 & 15 & \text{all} \: h_i \: \text{are equal} \\ \hline 5 & 15 & h_i \leq h_{i+1} \: \text{for all} \: i \leq N-1 \\ \hline 6 & 18 & N \leq 1000\\ \hline 7 & 27 & \text{no additional constraints}\\ \hline \end{array}\)
2 1 2 1 2
12
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:
