| Allison has just developed the latest addictive mobile game: **Super Path |
| Drawer: Extreme Edition**! |
|
|
| This game takes place on an infinite 2D plane (thanks to advancements in |
| graphical technology, the entire plane can fit onto a mobile phone screen). |
| Two distinct points on the plane are chosen: a starting point (at coordinates |
| (0, **S**)), and an ending point (at coordinates (1,000,000,000, **E**)). The |
| player's goal is to draw a path from the starting point to the ending point. |
| The path may be any continuous curve on the plane (not necessarily a straight |
| line segment), and may cross itself. If the player successfully draws such a |
| path, they're rewarded with a "point", thus convincing them that their time is |
| being put to good use. If they fail to draw such a path, they're instead |
| forced to watch an ad, thus motivating them to do better on their next |
| attempt. |
|
|
| By itself, this game is certainly addicting, but it doesn't seem to cause |
| players to watch enough ads! So, as a last-minute addition, Allison has |
| inserted some lasers. There are **N** laser emitters, the _i_th of which is at |
| (**Xi**, **Yi**). Each emitter will emit a laser beam either directly up, |
| down, left, or right. The laser beam is a ray starting from the emitter's |
| position (inclusive) and continuing infinitely in the selected direction. |
| Naturally, the player's path may not touch any part of any laser beam. All |
| **N**+2 x-coordinates (of the starting point, ending point, and laser |
| emitters) are distinct, and all **N**+2 y-coordinates are also distinct, |
| meaning that it's impossible for a laser beam to ever directly hit the |
| starting point, ending point, or another emitter. |
|
|
| Allison hasn't programmed in the capability for moving laser emitters around, |
| but she can at least cause them to emit their laser beams in different |
| combinations of directions each time a player replays the game. She's |
| concerned that players will get bored as soon as they encounter a laser |
| configuration which they've already seen, so she'll make sure that each player |
| is presented with each of the 4**N** possible different laser configurations |
| exactly once. |
|
|
| Some laser configurations result in the ending point still being reachable |
| from the starting point by some continuous path which doesn't touch any laser |
| beams, in which case players will surely manage to find such a path. But for |
| other laser configurations, no valid path exists at all, resulting in a forced |
| ad showcase. Allison would like to count the total number of ads which a |
| player will end up watching upon playing the game once for each of the 4**N** |
| possible different laser configurations. As this value can be large, you only |
| need to compute it modulo 1,000,000,007. |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of sets of lasers. For each set |
| of lasers, there is first a line containing the space-separated integers |
| **N**, **S**, and **E**. Then, **N** lines follow. The _i_th of these contains |
| the space-separated integers **Xi** and **Yi**. |
|
|
| ### Output |
|
|
| For the _i_th set of lasers, output a line containing "Case #_i_: " followed |
| by the number of ads shown to each player, modulo 1,000,000,007. |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 30 |
| 1 ≤ **N** ≤ 50 |
| 0 ≤ **S**, **E**, **Xi**, **Yi** ≤ 1,000,000,000 |
|
|
| ### Explanation of Sample |
|
|
| In the first case, 11 of the 64 possible laser configurations result in the |
| ending point being unreachable from the starting point. One such configuration |
| is when the first emitter points down, the second points up, and the third |
| points right. Another is when the first emitter points down, the second points |
| left, and the third points down. |
|
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|
