| Wilson, everyone's favorite recurring programming contest character, has |
| finally decided that it's time to retire. Why? After several more job changes |
| and miraculous promotions, not only has he become incredibly wealthy, but he's |
| ended up stuck in a boring administrative role, which was the last thing he |
| wanted! |
|
|
| Wilson is currently employed as the head of accounting at a pie delivery |
| company, and has been given one last assignment to complete before his |
| retirement — computing delivery truck gas usage. He wishes he could just drive |
| one of the trucks instead... |
|
|
| The pie delivery company services a network of **N** \+ 1 towns, which rather |
| looks like a pie itself (or perhaps like an umbrella...). **N** of the towns |
| form a circle around the outside of the region, and are numbered from 1 to |
| **N** in clockwise order. They are connected to one another by **N** roads, |
| each of which can be driven along in either direction. The _i_th of these |
| roads requires **Oi** litres of gas to drive along, and runs between towns _i_ |
| and _i_ \+ 1 (unless _i_ = **N**, in which case it instead runs between towns |
| **N** and 1). |
|
|
| The remaining town, numbered **N** \+ 1, lies in the center of the region. |
| There are **N** roads connecting it to the other towns, each of which can be |
| driven along in either direction. The _i_th of these roads requires **Ri** |
| litres of gas to drive along, and runs between towns **N** \+ 1 and _i_. |
|
|
| A total of **N** * (**N** \+ 1) / 2 pie deliveries are scheduled to be carried |
| out, one for each distinct pair of towns. In particular, for every town _i_, |
| _i_ \- 1 deliveries will originate from it, with the _j_th of them having town |
| _j_ as its destination. The truck drivers will have to pay the gas expenses |
| out of their own pockets, of course, so for each delivery, the truck driver |
| will choose a sequence of roads to follow from the initial town to the |
| destination such that the total amount of gas required for their trip is |
| minimized. Wilson's task is to tally up the total amount of gas which will be |
| used across all **N** * (**N** \+ 1) / 2 of the deliveries. He's been |
| instructed to only determine the value of this sum modulo 1,000,000,007. |
|
|
| You're given **O1**, and **O2..N** may then be calculated as follows using |
| given constants **Ao**, **Bo**, **Co**, and **Do**. |
|
|
| **Oi** = ((**Ao** * **Oi-1** \+ **Bo**) %**Co** \+ **Do** |
|
|
| You're given **R1**, and **R2..N** may then be calculated as follows using |
| given constants **Ar**, **Br**, **Cr**, and **Dr**. |
|
|
| **Ri** = ((**Ar** * **Ri-1** \+ **Br**) %**Cr** \+ **Dr** |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of different regions of towns. |
| For each region, there is first a line containing the integer **N**. Then |
| there is a line with five space-separated integers, **O1**, **Ao**, **Bo**, |
| **Co**, and **Do**. Then there is a line with five space-separated integers, |
| **R1**, **Ar**, **Br**, **Cr**, and **Dr**. |
|
|
| ### Output |
|
|
| For the _i_th graph, print a line containing "Case #**i**: " followed by the |
| total amount of gas which will be used (in litres), modulo 1,000,000,007 |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 40 |
| 3 ≤ **N** ≤ 1,000,000 |
| 1 ≤ **O1**, **Co**, **Do** ≤ 1,000,000 |
| 0 ≤ **Ao**, **Bo** ≤ 1,000,000 |
| 1 ≤ **R1**, **Cr**, **Dr** ≤ 1,000,000 |
| 0 ≤ **Ar**, **Br** ≤ 1,000,000 |
|
|
| ### Explanation of Sample |
|
|
| In the first case, the roads running around the outside require 1, 3, and 5 |
| litres of gas to drive along, respectively. Meanwhile, the roads connected to |
| the central town require 1, 2, and 2 litres of gas to drive along, |
| respectively. The amounts of gas required for the 6 deliveries (in litres) are |
| [1, 1, 2, 2, 3, 3], for a total of 12. |
|
|
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|
