| Some time has passed since you took in the local population of snakes, feeding | |
| them every day and allowing them to live peacefully amongst your ladders. | |
| Unfortunately, your neighbours have not been thrilled about living next to | |
| billions of serpents, so they've filed complaints to the mayor! As a result, | |
| almost all of your snakes and ladders have been forcibly evicted, leaving you | |
| with only one of each for company. | |
| Your ladder has **N** horizontal rungs, numbered 1 to **N** from bottom to | |
| top. Like all ladders, it also has 2 vertical rails, with rail 1 on the left | |
| and rail 2 on the right. To improve its appearance, you've allowed some plants | |
| to grow all over it. Notably, there are **K** flowers at distinct locations on | |
| the ladder, with the ith flower at the intersection of rung **R_i** and rail | |
| **C_i**. | |
| Your snake loves to sleep on the ladder. It always places its head at the | |
| intersection of some rung and some rail, and from there, each subsequent | |
| segment of its body extends up, down, left, or right to follow a rung or rail | |
| to another intersection. The snake likes to be spread out, but hates touching | |
| flowers, so it insists that the path that its body takes must not touch any | |
| intersection more than once, and must not touch any intersection that has a | |
| flower. Conveniently, the snake is exactly long enough to theoretically pass | |
| through all intersections that don't have flowers — that is, if consecutive | |
| rungs and rails are 1 unit apart from one another, then the snake has a length | |
| of 2 * **N** \- **K** \- 1 units. Note that your snake may have 0 length (in | |
| which case it'll only occupy a single intersection). | |
| The diagram below (which corresponds to the fourth sample case) illustrates a | |
| valid way in which the snake can arrange itself on a ladder with 4 rungs and 1 | |
| flower: | |
|  | |
| Can you help your snake count the number of different ways in which it can | |
| position itself on the ladder, such that it covers all of the intersections | |
| which don't contain flowers? An arrangement of the snake is defined by an | |
| ordered sequence of intersections that it passes through, starting from its | |
| head. You only need to compute the answer modulo 109 \+ 7. | |
| ### Input | |
| Input begins with an integer **T**, the number of ladders. For each ladder, | |
| there is first a line containing the space-separated integers **N** and **K**. | |
| Then **K** lines follow, the **i**th of which contains the space-separated | |
| integers **Ri** and **Ci** . | |
| ### Output | |
| For the **i**th ladder, print a line containing "Case #**i**: " followed by | |
| the number of ways in which the snake can be arranged, modulo 109 \+ 7. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 555 | |
| 1 ≤ **N** ≤ 1012 | |
| 0 ≤ **K** ≤ min(1000, 2 * **N** \- 1) | |
| 1 ≤ **Ri** ≤ **N** | |
| 1 ≤ **Ci** ≤ 2 | |
| ### Explanation of Sample | |
| For the first ladder, the snake can place its head in any of the 4 | |
| intersections, and from each one it can arrange itself in 2 ways for 8 ways | |
| total. For the second ladder, the flowers prevent the snake from crossing from | |
| the top 2 intersections to the bottom 2 (or vice versa). For the third ladder, | |
| there are 2 possible intersections at which the snake can place its head which | |
| yield 1 valid arrangement each. | |