| You live on a circular road, **L** metres in length. Any point on the road can |
| be referred to by a real number **x** (0 ≤ **x** < **L**), the distance along |
| the road clockwise from its Northmost point to that point (in metres). |
|
|
| Fortunately for you, this road is served by public transportation! There are |
| **N** bus stops at distinct, integral points along the road. |
|
|
| Unfortunately for you, due to budget cuts exactly **K** of these **N** stops |
| will soon be removed. The group of **K** removed stops will be chosen |
| uniformly at random from the set of all possible groups of **K** stops. |
|
|
| You'd like to calculate the expected distance you'll have to walk from a |
| random point along the road, chosen uniformly at random from the interval [0, |
| **L**), to the nearest of the remaining **N** \- **K** bus stops, in metres. |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of roads. For each road, there |
| is first a line containing the space-separated integers **N**, **K**, and |
| **L**. Then follows a line containing a string of length **L**. This string |
| consists of only the characters '0' and '1'. There is a bus stop at position |
| **x** if and only if the (**x** \+ 1)th character of the string is '1'. |
| Exactly **N** of the characters will be '1'. |
|
|
| ### Output |
|
|
| For the **i**th road, print a line containing "Case #**i**: " followed by the |
| expected distance you'll have to walk from a random point to the nearest bus |
| stop, in metres. You should output the exact answer modulo (109 \+ 7). That |
| is, if the exact answer is **a** / **b** (where **a** and **b** are integers), |
| you should output **a** * **b**-1 mod (109 \+ 7) (where **b**-1 is the modular |
| inverse of **b** mod (109 \+ 7)). |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 20 |
| 1 ≤ **N** ≤ 500,000 |
| 0 ≤ **K** < **N** |
| 1 ≤ **L** ≤ 1,000,000 |
|
|
| ### Explanation of Sample |
|
|
| In the first case, the single existing stop will remain untouched. If your |
| starting position is smaller than 1 or greater than 5, you'll walk clockwise |
| to it, for a distance of between 0m and 4m. Otherwise, you'll walk |
| counterclockwise to it, also for a distance of between 0m and 4m. As such, |
| your expected distance to walk will be 2m. |
|
|
| In the third case, one of the stops will be removed at random. Whichever one |
| is removed, your situation will be similar to that of the first case - the |
| distance you'll have to walk will be uniformly distributed between 0m and 4m, |
| for an expected distance of 2m. |
|
|
| In the fourth case, the exact answer is 3.15 or 63/20, which is 550000007 when |
| taken modulo (109 \+ 7). |
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|
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|
