Harry Potter is hiking in the Alps surrounding Lake Geneva. In this area there are $$$m$$$ cabins, numbered 1 to $$$m$$$. Each cabin is connected, with one or more trails, to a central meeting point next to the lake. Each trail is either short or long. Cabin $$$i$$$ is connected with $$$s_i$$$ short trails and $$$l_i$$$ long trails to the lake.
Each day, Harry walks a trail from the cabin where he currently is to Lake Geneva, and then from there he walks a trail to any of the $$$m$$$ cabins (including the one he started in). However, as he has to finish the hike in a day, at least one of the two trails has to be short.
How many possible combinations of trails can Harry take if he starts in cabin 1 and walks for $$$n$$$ days?
Give the answer modulo $$$10^9 + 7$$$.
The first line contains the integers $$$m$$$ and $$$n$$$.
The second line contains $$$m$$$ integers, $$$s_1, \dots, s_m$$$, where $$$s_i$$$ is the number of short trails between cabin $$$i$$$ and Lake Geneva.
The third and last line contains $$$m$$$ integers, $$$l_1, \dots, l_m$$$, where $$$l_i$$$ is the number of long trails between cabin $$$i$$$ and Lake Geneva.
We have the following constraints:
$$$0 \le s_i, l_i \le 10^3$$$.
$$$1 \le m \le 10^5$$$.
$$$1 \le n \le 10^9$$$.
The number of possible combinations of trails, modulo $$$10^9 + 7$$$.
3 21 0 10 1 1
18
