| A quiet evening has set over a residential area. As families sit down for |
| supper in the safety of their homes, a calm atmosphere permeates the outside |
| air. The neighborhood feels truly at peace, separated from the frenzy of the |
| rest of the world. Also, a bunch of zombies have just risen out of the ground |
| and want to eat everybody. |
|
|
| The neighborhood has **N** yards in a row, numbered from 1 to **N**. There are |
| also **N**-1 fences, one between each pair of adjacent yards. The fence |
| between yards _i_ and _i_+1 has an unknown integral height drawn uniformly at |
| random from the inclusive interval [**Ai**, **Bi**]. In other words, the _i_th |
| fence has **Bi** \- **Ai** \+ 1 possible heights, each of which is equally |
| likely. |
|
|
| **M** hungry zombies are also present, with the _i_th of them initially in yard **Yi**. Fortunately for the zombies, they might not be stopped by the surrounding fences so easily. The _i_th zombie has the ability to climb over any fence with a height of at most **Hi**. It may repeatedly move from its current yard to an adjacent one, as long as the fence between the yards is no taller than **Hi**. Multiple zombies may start in the same yard, and multiple zombies may occupy the same yard at any point. |
|
|
| A yard is considered "safe" if it's impossible for any zombies to ever reach |
| it. Determine the probability that at least one of the **N** yards is safe. |
| Let this probability be represented as a quotient of integers _p_/_q_ in |
| lowest terms. Output the value of this quotient modulo 1,000,000,007 — in |
| other words, output the unique integer _x_ such that 0 ≤ _x_ < 1,000,000,007 |
| and _p_ = _x_*_q_ (modulo 1,000,000,007). |
| |
| ### Input |
| |
| Input begins with an integer **T**, the number of neighborhoods. For each |
| neighborhood, there is first a line containing the space-separated integers |
| **N** and **M**. Then, **N-1** lines follow. The _i_th of these lines contains |
| the space-separated integers **Ai** and **Bi**. Then, **M** lines follow. The |
| _i_th of these lines contains the space-separated integers **Yi** and **Hi**. |
| |
| ### Output |
| |
| For the _i_th neighborhood, print a line containing "Case #_i_: " followed by |
| 1 integer, the probability that at least one of the yards is safe, expressed |
| as a quotient of integers modulo 1,000,000,007. |
| |
| ### Constraints |
| |
| 1 ≤ **T** ≤ 75 |
| 1 ≤ **N** ≤ 3,000 |
| 1 ≤ **M** ≤ 3,000 |
| 1 ≤ **Ai** ≤ **Bi** ≤ 1,000,000 |
| 1 ≤ **Yi** ≤ **N** |
| 1 ≤ **Hi** ≤ 1,000,000 |
| |
| ### Explanation of Sample |
| |
| In the first case, if the height of the single fence is 100, then the zombie |
| in yard 1 will be able to climb over it to reach yard 2, meaning that no yards |
| will be safe. Otherwise, if the fence's height is 101, then yard 2 will be |
| safe. Therefore, the probability that at least one of the yards is safe is 1/2 |
| = 500000004 (modulo 1,000,000,007). |
| |
| In the second case, in order for yard 2 to be safe from both surrounding |
| zombies, the first fence's height must be either 3 or 4, and the second |
| fence's height must be 4. The probability of this occurring is 2/4 * 1/4 = 1/8 |
| = 125000001 (modulo 1,000,000,007). |
|
|
| In the third case, the probability of at least one yard being safe is 2/3 = |
| 666666672 (modulo 1,000,000,007). |
|
|
|
|
