| **N** groups of people are heading to the beach today! The _i_th group is bringing a circular umbrella with a radius **Ri** meters. | |
| The beach has **M** acceptable points at which umbrellas may be screwed into | |
| the sand, arranged in a line with 1 meter between each adjacent pair of | |
| points. Each group of people will choose one such point at which to position | |
| the center of their umbrella. | |
| Of course, it's no good if any pair of umbrellas collide (that is, if the | |
| intersection of their circles has a positive area). The **N** groups will work | |
| together to place their umbrellas such that this doesn't happen. However, | |
| they're wondering in how many distinct ways that can be accomplished. Two | |
| arrangements are considered to be distinct if they involve at least one group | |
| placing their umbrella in a different spot. As this quantity may be very | |
| large, they're only interested in its value modulo 1,000,000,007. | |
| Note that it might be impossible for all of the groups to validly place their | |
| umbrellas, yielding an answer of 0. | |
| ### Input | |
| Input begins with an integer **T**, the number of days the beach is open. For | |
| each day, there is first a line containing two space-separated integers, **N** | |
| and **M**. Then, **N** lines follow, the _i_th of which contains a single | |
| integer, **Ri**. | |
| ### Output | |
| For the _i_th day, print a line containing "Case #**i**: " followed by the | |
| number of valid umbrella arrangements, modulo 1,000,000,007. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 100 | |
| 1 ≤ **N** ≤ 2,000 | |
| 1 ≤ **M** ≤ 1,000,000,000 | |
| 1 ≤ **Ri** ≤ 2,000 | |
| ### Explanation of Sample | |
| In the second case there are six possibilities. If the radius-1 umbrella is | |
| placed at the far-left point, then the radius-2 umbrella can be placed at | |
| either of the two right-most points. If the radius-1 umbrella is placed at the | |
| second point from the left, then the radius-2 umbrella must be placed at the | |
| right-most point. That's three possibilities so far, and we can mirror them to | |
| produce three more. | |