| Once upon a time, you were the proud owner of a great many ladders and snakes. | |
| Unfortunately, you were forced to give up all but one of each, and in time, | |
| even your single remaining snake slithered away... | |
| But all of that is about to change. You've just received word that a new | |
| executive order has been passed which will allow you to once again keep as | |
| many snakes as you'd like! To prepare, you've eagerly gone ahead and | |
| constructed a series of **N** ladders which will serve as a home for the | |
| impending flock of snakes. Unfortunately, it was only then that you realized | |
| your huge mistake — feeding snakes is extraordinarily expensive! | |
| The **N** ladders are arranged in a line on the ground, with each one standing | |
| up vertically. Each pair of consecutive ladders are 1 metre apart from each | |
| other, and the _i_th ladder from the left initially has a height of **Hi** | |
| metres. As an expert in reptilian behavioral patterns, you're sure that a | |
| certain number of snakes will soon arrive on your property. In particular, | |
| every possible unordered pair of ladders will surely be claimed by a single | |
| snake, meaning that exactly **N** * (**N** \- 1) / 2 snakes will be showing | |
| up. If a snake claims the pair of ladders _i_ and _j_, it will want to stretch | |
| itself out perfectly between the tops of those two ladders, such that its body | |
| runs down one ladder, along the ground, and up along the other ladder. | |
| Therefore, such a snake will surely have a length of exactly **Hi** \+ |_j_ \- | |
| _i_| + **Hj** metres. | |
| You're desperate to reduce the heights of some of your ladders as quickly as | |
| possible so as to attract some shorter snakes and save your wallet. You | |
| estimate that you've got **K** minutes to make your alterations before the | |
| snakes start showing up. Each minute, you may choose a single ladder and cut | |
| off its top few rungs, reducing its height by exactly 1 metre. You may not | |
| shorten a ladder if it's already only 1 metre tall. You may choose not to cut | |
| any ladders in a given minute. | |
| As everyone knows, the daily cost of feeding a snake is proportional to its | |
| length. That being said, you're not concerned with the total amount you'll | |
| have to dish out every day, but rather on the largest amount you'll have to | |
| spend on any one snake. As such, you're like to determine the minimum possible | |
| length that the _longest_ of the **N** * (**N** \- 1) / 2 snakes can end up | |
| having, given that you perform your ladder cutting optimally. | |
| You're given **H1**, and **H2..N** may then be calculated as follows using | |
| given constants **A**, **B**, and **C**. | |
| **Hi** = ((**A** * **Hi-1** \+ **B**) %**C** \+ 1 | |
| ### Input | |
| Input begins with an integer **T**, the number of different sets of ladders. | |
| For each set of ladders, there is first a line containing the space-separated | |
| integers **N** and **K**. Then there is a line with four space-separated | |
| integers, **H1**, **A**, **B** and **C**. | |
| ### Output | |
| For the _i_th snake, print a line containing "Case #**i**: " followed by the | |
| minimum possible length that the longest snake can have (in metres). | |
| ### Constraints | |
| 1 ≤ **T** ≤ 20 | |
| 2 ≤ **N** ≤ 800,000 | |
| 0 ≤ **K** ≤ 1015 | |
| 1 ≤ **H1**, **C** ≤ 1,000,000,000 | |
| 0 ≤ **A**, **B** ≤ 1,000,000,000 | |
| ### Explanation of Sample | |
| In the first case the ladders have heights of 4, 5, and 6 metres respectively. | |
| Cutting the last ladder 3 times will yields height of 4, 5, and 3. The longest | |
| snake is then the one which claims the first two ladders, with a length of 4 + | |
| 1 + 5 = 10 metres. You could instead cut the second ladder once and the third | |
| ladder twice for final heights of 4, 4, and 4. In this case, the longest snake | |
| is the one which claims the first and last ladders, with a length of 4 + 2 + 4 | |
| = 10 metres. | |
| In the second case, you have more than enough time to trim all of the ladders | |
| down to a height of 1 metre each. The longest snake is then the one which | |
| claims the first and last ladders, with a length of 1 + 4 + 1 = 6 metres. | |
| In the third case, the ladder heights are [6, 16, 36, 2, 8, 20, 7, 18]. | |