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).