| Uh oh, the weather forecast predicts that a rainstorm will soon break out over | |
| the infinite, 1-dimensional number line you call home! | |
| The forecast is remarkably precise. In fact, it's known that exactly **N** | |
| raindrops will fall, with the **i**th drop striking the number line at | |
| position **Xi** exactly **Ki** seconds after the start of the storm. No two | |
| drops will strike the number line at exactly the same position and time. | |
| You'd like to stop as many of the drops as possible, but you don't exactly | |
| have an umbrella... so your boomerang will have to do. Your plan is as | |
| follows: | |
| 1. Stand at some (possibly non-integral) position A on the number line, and choose some other position B (A ≠ B). | |
| 2. At some point in time, throw your boomerang from A to B. This point in time can be arbitrarily long before or after the start of the storm (the storm won't start for a while, so you have time to prepare). The boomerang will travel along your chosen line segment at a constant speed of S units per second. | |
| 3. Make your boomerang spin in place at position B for some non-negative amount of time (which can be arbitrarily large). | |
| 4. Have your boomerang travel back along the line segment from B to A at the same speed, and catch it as soon as it gets back to position A. | |
| During the inclusive time interval from when you throw the boomerang to when | |
| you catch it again, if the boomerang is ever at exactly the same position as a | |
| raindrop at exactly the time that it strikes the number line, it will | |
| intercept it... just like an umbrella! What's the maximum number of raindrops | |
| you can intercept in this way with a single throw? | |
| ### Input | |
| Input begins with an integer **T**, the number of rainstorms. For each | |
| rainstorm, there is first a line containing the space-separated integers **N** | |
| and **S**. Then, **N** lines follow, the **i**th of which contains the space- | |
| separated integers **Xi** and **Ki**. | |
| ### Output | |
| For the **i**th rainstorm, print a line containing "Case #**i**: " followed by | |
| the maximum number of raindrops you can stop. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 50 | |
| 1 ≤ **N** ≤ 100,000 | |
| 1 ≤ **S** ≤ 1,000,000,000 | |
| 0 ≤ **Xi**, **Ki** ≤ 1,000,000,000 | |
| ### Explanation of Sample | |
| In the first rainstorm, one solution is to stand at X = 99 and let your | |
| boomerang spin at X = 100 until it intercepts all three raindrops that will | |
| fall at that position. In the second rainstorm, one solution is to stand at X | |
| = 50 and throw your boomerang towards X = 100 as soon as the first raindrop | |
| hits it. If you recall your boomerang as soon as it reaches X = 100, it will | |
| hit the second raindrop and be back just in time to hit the third. | |