| Today you've found yourself standing on an infinite 2D plane at coordinates | |
| (**X0**, **Y0**). There are also **N** targets on this plane, with the **i**th | |
| one at coordinates (**Xi**, **Yi**). | |
| You have a boomerang which you can throw in a straight line in any direction | |
| from your initial location. After you throw it, you may instantaneously run to | |
| any location on the plane. After the boomerang has travelled a distance of | |
| exactly **D** along its initial trajectory, it will return directly to you — | |
| that is, to your chosen final location. Note that you cannot move around once | |
| the boomerang has started its return trip — its path will always consist of 2 | |
| line segments (the first of which has a length of exactly **D**). The | |
| boomerang and the targets have infinitesimal size. | |
| Let **A** be the number of targets which your boomerang hits (directly passes | |
| through) during the first segment of its flight, and **B** be the number of | |
| targets which it hits during the second segment. Your throw is then awarded a | |
| score of **A** * **B**. What's the maximum score you can achieve? Note that, | |
| if there is a target at the exact location at which the two segments meet (at | |
| a distance of **D** from your initial location), then it counts towards both | |
| **A** and **B**! | |
| ### Input | |
| Input begins with an integer **T**, the number of planes. For each plane, | |
| there is first a line containing the space-separated integers **X0** and | |
| **Y0**. The next line contains the integer **D**, and the one after contains | |
| the integer **N**. Then, **N** lines follow, the **i**th of which contains the | |
| space-separated integers **Xi** and **Yi**. | |
| ### Output | |
| For the **i**th plane, print a line containing "Case #**i**: " followed by the | |
| maximum score you can achieve. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 20 | |
| 1 ≤ **N** ≤ 3,000 | |
| 1 ≤ **D** ≤ 100 | |
| -100 ≤ **Xi**, **Yi** ≤ 100, for 0 ≤ **i** ≤ **N** | |
| All coordinates are pairwise distinct. The following restrictions are also | |
| guaranteed to hold for the input given: | |
| For any three targets at distinct points **a**, **b**, and **c**, it is | |
| guaranteed that **c** is either closer than 10-13 away from the infinite line | |
| between **a** and **b** (and is considered to be on the line), or is further | |
| than 10-6 away (and is considered to not be on the line). | |
| Let **p** be any point at which the boomerang may change direction after | |
| hitting a target. For any two targets at distinct points **a** and **b**, it | |
| is guaranteed that **p** is either closer than 10-13 away from the infinite | |
| line between **a** and **b** (and is considered to be on the line), or is | |
| further than 10-6 away (and is considered to not be on the line). | |
| ### Explanation of Sample | |
| On the first plane, one optimal strategy is to throw the boomerang in the | |
| direction of the positive x-axis (that is, to (6, 0)), and then run to (0, 0). | |
| It will hit targets 2 and 3 on the first segment of its flight, and all 3 | |
| targets on the second segment, for a score of 2*3=6. | |