| <p> | |
| A rich, retired programmer has decided to invest his (possibly) massive savings into constructing custom yachts for himself. | |
| </p> | |
| <p> | |
| Initially, he has a real amount of dollars, chosen uniformly at random from the range [<strong>A</strong>, <strong>B</strong>]. | |
| Constructing a yacht consists of <strong>N</strong> sequential steps, with the <strong>i</strong>th step requiring <strong>C<sub>i</sub></strong> dollars. | |
| </p> | |
| <p> | |
| The programmer blindly pays for the steps in order, until he's either completed all of them, or can't afford the cost of the next step. | |
| If the former occurs, he puts his completed yacht aside and restarts the process from the first step with his remaining money - he wants as many yachts as possible! | |
| Otherwise, in the latter case, he immediately stops his project entirely, without spending any additional money on other steps. | |
| </p> | |
| <p> | |
| What's the expected amount of money which the programmer will be left with once he stops spending it on yachts? Your output should have at most 10<sup>-6</sup> absolute or relative error. | |
| </p> | |
| <h3>Input</h3> | |
| <p> | |
| Input begins with an integer <strong>T</strong>, the number of times the programmer embarks on a yacht-creation spree. | |
| For each spree, there is first a line containing the space-separated integers | |
| <strong>N</strong>, <strong>A</strong>, and <strong>B</strong> in that order, | |
| then a line containing <strong>N</strong> | |
| space-separated integers, the <strong>i</strong>th of which is <strong>C<sub>i</sub></strong>. | |
| </p> | |
| <h3>Output</h3> | |
| <p> | |
| For the <strong>i</strong>th spree, print a line containing "Case #<strong>i</strong>: " followed by | |
| the expected amount of money the programmer will have left. | |
| </p> | |
| <h3>Constraints</h3> | |
| <p> | |
| 1 ≤ <strong>T</strong> ≤ 50 <br /> | |
| 0 ≤ <strong>A</strong> < <strong>B</strong> ≤ 1,000,000,000 <br/> | |
| 1 ≤ <strong>N</strong> ≤ 100,000 <br /> | |
| 1 ≤ <strong>C<sub>i</sub></strong> ≤ 1,000,000,000 <br /> | |
| </p> | |
| <h3>Explanation of Sample</h3> | |
| <p> | |
| In the first case, the programmer starts with between 5 and 8 dollars. The programmer's initial amount of money will fall into one of the ranges [5, 6), or [6, 8), or [8, 8]. | |
| If it falls into the first range, the programmer will end with [1, 2) dollars. If it falls into the second range, the programmer will end with [0, 2) dollars. If the programmer | |
| starts with exactly 8 dollars, he'll end with 0 dollars. | |
| </p> | |