| It's time for your school class to elect its class officers for the year. | |
| These officers will have the honour of representing the class and will hold | |
| various responsibilities. The position of class treasurer is particularly | |
| appealing to some of the more organized and mathematically-inclined students, | |
| especially two rivals, Amy and Betty. They've both been running strong | |
| campaigns, to the point that no other feasible candidates remain. | |
| A democratic vote to elect the class treasurer is about to take place! Each of | |
| the **N** students in the class will cast one vote, for either Amy or Betty. | |
| The students have IDs numbered from 1 to **N**, and student _i_ is currently | |
| planning on voting for either Amy (if **Vi** = "A") or Betty (if **Vi** = | |
| "B"). | |
| Your teacher, Mr. X, has some "novel" ideas about how elections should work. | |
| Perhaps in an effort to teach your class a statistics lesson, he will organize | |
| the election as follows. He'll consider the set of all **N***(**N**+1)/2 | |
| possible non-empty contiguous sets of student IDs, and will select one at | |
| random. Each set will have an equal chance of being selected. He'll refer to | |
| this as the "representative set" of students. He will also announce a | |
| threshold of victory, **K**. He'll then tally up the number of votes for Amy | |
| and Betty amongst those students — let these vote counts be **a** and **b** | |
| respectively. If **a** > **b** \+ **K**, then Amy will win. If **b** > **a** | |
| \+ **K**, then Betty will win. Otherwise, if |**a** \- **b**| ≤ **K**, then it | |
| will be a draw (neither candidate will win, and the class will be left without | |
| a treasurer). | |
| Amy has grown concerned about how the election will turn out, so she's | |
| enlisted your help in potentially swaying some of your classmates' opinions. | |
| She's tasked you with ensuring that, no matter which representative set gets | |
| chosen, Betty cannot possibly win (in other words, either Amy will win or | |
| neither candidate will win). To do so, you may pay 0 or more students to | |
| change their vote from their current candidate to the other one. Student _i_ | |
| requires 2i dollars to be influenced in this fashion. You must finish paying | |
| students off _before_ you know what the representative set will be. | |
| What's the minimum possible cost required to guarantee that Betty cannot | |
| possibly win and become the class treasurer? As this cost may be large, output | |
| it modulo 1,000,000,007. Note that you must minimize the actual cost, rather | |
| than minimizing the resulting value of the cost after it's taken modulo | |
| 1,000,000,007. | |
| ### Input | |
| Input begins with an integer **T**, the number of elections. For each | |
| election, there are two lines. The first line contains the space-separated | |
| integers **N** and **K**. The second line contains the **N** characters **V1** | |
| through **VN**. | |
| ### Output | |
| For the _i_th election, print a line containing "Case #_i_: " followed by 1 | |
| integer, the minimum possible cost (in dollars) required to guarantee that | |
| Betty cannot become the class treasurer, modulo 1,000,000,007. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 200 | |
| 1 ≤ **N** ≤ 1,000,000 | |
| 0 ≤ **K** ≤ **N** | |
| ### Explanation of Sample | |
| In the first case, if you do nothing, then Betty may win (if the | |
| representative set either consists of only student 1 or only student 4). You | |
| should pay students 1 and 4 to each vote for Amy instead of Betty, for a cost | |
| of $2 + $16 = $18. Amy will then be guaranteed to win for any choice of | |
| representative set. | |
| In the second case, you don't need to pay any students to change their minds — | |
| either Amy will win, or it will be a draw. | |
| In the third case, you should pay student 2 $4 to change their vote to Amy. | |