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.