| A certain forest has **N** trees growing in it, and there just so happens to |
| be a Fox living at the top of each one! The trees are numbered from 1 to |
| **N**, and their bases are all arranged in a straight line on the ground, with |
| 1 metre between the bases of each pair of adjacent trees _i_ and _i_ \+ 1. |
| Each tree _i_ is **Hi** metres tall. |
|
|
| The Foxen are all good friends with one another, and frequently like to go out |
| for strolls to visit each other's homes. Rather than jumping directly between |
| the trees, they prefer to always keep their paws firmly planted on some |
| surface, such as tree trunks or the ground. As such, the shortest possible |
| trip from the top of tree _i_ to the top of another tree _j_ requires climbing |
| down tree _i_ to the ground, walking along the ground to the base of tree _j_, |
| and then climbing up to its top, resulting in a total distance of **Hi** \+ |
| |_j_ \- _i_| + **Hj** metres traveled. |
|
|
| However, the Foxen aren't terribly satisfied with how long their trips |
| currently take. Given the frequency of their strolls, they've decided to |
| invest in reducing their travel distance by constructing some bridges between |
| the tree trunks. They're only interested in building metre-long, horizontal |
| bridges. Each bridge may be constructed to connect any pair of adjacent trees |
| _i_ and _i_ \+ 1, and it may be placed at any height above the ground, as long |
| as it touches both of those tree trunks (in other words, its height must be no |
| larger than the minimum of **Hi** and **Hi+1**). Multiple bridges may be |
| constructed at different heights between any given pair of adjacent trees. |
| Once bridges are installed, the Foxen will be willing to walk across them, |
| potentially saving them the time of descending all the way to the ground |
| during their strolls. |
|
|
| There are **N** * **(N - 1)** / 2 different pairs of Foxen _i_ and _j_ who |
| might want to meet up, requiring a trip to be taken from the top of tree _i_ |
| to the top of tree _j_ (or vice versa). The Foxen's top priority is minimizing |
| the sum of the **N** * **(N - 1)** / 2 pairwise shortest distances of strolls |
| which would be required for these visits to take place. Please help them |
| determine the minimum possible value of this sum, assuming that they construct |
| as many bridges as it takes. That being said, they don't want to spend all day |
| constructing bridges either... as such, they're also interested in the minimum |
| number of bridges which they can construct to achieve that minimum possible |
| sum of pairwise distances. |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of forests. For each forest, |
| there are two lines. The first line contains the integer **N**. The second |
| line contains **N** space-separated integers, the _i_th of which is the |
| integer **Hi**. |
|
|
| ### Output |
|
|
| For the _i_th forest, print a line containing "Case #**i**: " followed by two |
| integers: the minimum possible sum of pairwise shortest distances after any |
| number of bridges are built (in metres), and the minimum number of bridges |
| required to achieve that minimum sum. |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 30 |
| 2 ≤ **N** ≤ 500,000 |
| 1 ≤ **Hi** ≤ 10,000,000 |
| The sum of **N** values across all **T** cases does not exceed 6,000,000. |
|
|
| ### Explanation of Sample |
|
|
| In the first case, the Foxen's optimal strategy is to construct one bridge |
| between the first 2 trees at a height of 20m, and another bridge between the |
| last 2 trees at a height of 10m. With this configuration, the shortest path |
| between the first 2 trees' tops is 11m long, as is the shortest path for the |
| last 2 trees. The shortest path between the first tree's top and the last |
| tree's top is 22m long. The resulting sum of pairwise shortest distances is |
| then 44m, which is the minimum possible sum. |
|
|
|
|
