<p> Today you've found yourself standing on an infinite 2D plane at coordinates (<strong>X<sub>0</sub></strong>, <strong>Y<sub>0</sub></strong>). There are also <strong>N</strong> targets on this plane, with the <strong>i</strong>th one at coordinates (<strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong>). </p> <p> 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 <strong>D</strong> 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 <strong>D</strong>). The boomerang and the targets have infinitesimal size. </p> <p> Let <strong>A</strong> be the number of targets which your boomerang hits (directly passes through) during the first segment of its flight, and <strong>B</strong> be the number of targets which it hits during the second segment. Your throw is then awarded a score of <strong>A</strong> * <strong>B</strong>. 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 <strong>D</strong> from your initial location), then it counts towards both <strong>A</strong> and <strong>B</strong>! </p> <h3>Input</h3> <p> Input begins with an integer <strong>T</strong>, the number of planes. For each plane, there is first a line containing the space-separated integers <strong>X<sub>0</sub></strong> and <strong>Y<sub>0</sub></strong>. The next line contains the integer <strong>D</strong>, and the one after contains the integer <strong>N</strong>. Then, <strong>N</strong> lines follow, the <strong>i</strong>th of which contains the space-separated integers <strong>X<sub>i</sub></strong> and <strong>Y<sub>i</sub></strong>. </p> <h3>Output</h3> <p> For the <strong>i</strong>th plane, print a line containing "Case #<strong>i</strong>: " followed by the maximum score you can achieve. </p> <h3>Constraints</h3> <p> 1 ≤ <strong>T</strong> ≤ 20 <br /> 1 ≤ <strong>N</strong> ≤ 3,000 <br /> 1 ≤ <strong>D</strong> ≤ 100 <br /> -100 ≤ <strong>X<sub>i</sub></strong>, <strong>Y<sub>i</sub></strong> ≤ 100, for 0 ≤ <strong>i</strong> ≤ <strong>N</strong> <br /> </p> <p> All coordinates are pairwise distinct. The following restrictions are also guaranteed to hold for the input given: </p> <p> For any three targets at distinct points <strong>a</strong>, <strong>b</strong>, and <strong>c</strong>, it is guaranteed that <strong>c</strong> is either closer than 10<sup>-13</sup> away from the infinite line between <strong>a</strong> and <strong>b</strong> (and is considered to be on the line), or is further than 10<sup>-6</sup> away (and is considered to not be on the line). </p> <p> Let <strong>p</strong> be any point at which the boomerang may change direction after hitting a target. For any two targets at distinct points <strong>a</strong> and <strong>b</strong>, it is guaranteed that <strong>p</strong> is either closer than 10<sup>-13</sup> away from the infinite line between <strong>a</strong> and <strong>b</strong> (and is considered to be on the line), or is further than 10<sup>-6</sup> away (and is considered to not be on the line). </p> <h3>Explanation of Sample</h3> <p> 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. </p>