| A certain well-hidden valley is home to a thriving population of mysterious |
| creatures — Foxen! However, keeping the valley safe from outsiders (such as |
| humans) is a necessity. To that end, a group of Foxen have been sent out to |
| patrol the border. |
|
|
| On their patrol route, the Foxen know that they're going to pass by an |
| interesting, rectangular forest. When viewed from above, the forest can be |
| modeled as a grid of cells with **R** rows and **C** columns. The rows are |
| numbered from 1 to **R** from North to South, while the column are numbered |
| from 1 to **C** from West to East. One tree is growing in the center of each |
| cell, and each tree's height (in metres) is some positive integer no larger |
| than **H**. |
|
|
| If the Foxen were to look at the forest from the North side, all of the trees |
| in any given column of cells would obscure each other and blend together. In |
| fact, the Foxen would really only be able make out the overall shape of the |
| forest's "skyline" when viewed from that direction. This Northern skyline can |
| be expressed as a sequence of **C** positive integers, with the _i_th one |
| being the largest of the **R** tree heights in the _i_th column. |
|
|
| Similarly, if they were to look at the forest from the West side, they would |
| only be able to make out the shape of its skyline from that direction. This |
| Western skyline is a sequence of **R** positive integers, with the _i_th one |
| being the largest of the **C** tree heights in the _i_th row. |
|
|
| On their way to the forest, the Foxen find themselves wondering about what it |
| might look like. They've done their research and are aware of its dimensions |
| **R** and **C**, as well as the maximum possible height of its trees **H**, |
| but they don't know the actual heights of any of its trees. They'd like to |
| determine how many different, distinct-looking forests they might end up |
| finding. A forest is a set of heights for all **R**x**C** trees, and two |
| forests are considered to be distinct-looking from one another if their |
| Northern skyline sequences differ and/or their Western skyline sequences |
| differ. |
|
|
| Please help the Foxen determine the number of possible different, distinct- |
| looking forests! As this quantity may be quite large, they're only interested |
| in its value when taken modulo 1,000,000,007. |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of different forests visited by |
| the Foxen. For each forest, there is a single line containing the three space- |
| separated integers **R**, **C**, and **H**. |
|
|
| ### Output |
|
|
| For the _i_th forest, print a line containing "Case #**i**: " followed by the |
| number of possible different, distinct-looking forests modulo 1,000,000,007. |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 30 |
| 1 ≤ **R**, **C**, **H** ≤ 500,000 |
|
|
| ### Explanation of Sample |
|
|
| In the first case, there are 10 possible different, distinct-looking forests |
| which consist of a 2x2 grid of trees, with each tree being either 1m or 2m |
| tall. For example, the following 2 forests look different (even though their |
| Western skylines are equal, their Northern skylines differ), so both should be |
| counted: |
|
|
| 1 2 2 1 |
| 1 1 1 1 |
| |
| On the other hand, the following 2 forests look identical to one another from |
| both the North and the West, so only one of them should be counted: |
|
|
| 1 2 2 2 |
| 2 1 2 1 |
| |
|
|
