| Come one, come all! The most famous circus troupe in all the land is on tour! | |
| Having just arrived in a new location, they're eager to set up their big top | |
| and showcase world-class acts of acrobatics and comedy for their audience. | |
| Looking at the tent from the side (as a cross section), it will be set up | |
| along a one-dimensional strip of ground. **N** vertical poles will be placed | |
| in the ground, one after another, with the _i_th pole at a position **Xi** | |
| meters to the right of an arbitrary reference point, and reaching a height of | |
| **Hi** meters. No two poles will be at the same position. | |
| After each pole is placed, the shape of the tent will be updated to fit the | |
| current set of poles. In particular, the upper outline of the tent will be a | |
| function with the following properties: | |
| 1. it's defined over all positions from negative infinity to positive infinity | |
| 2. its height is always non-negative | |
| 3. it's made up entirely of a series of connected line segments, with each one having a slope with absolute value no greater than 1 (meaning that the function is continuous, and its height may never change from left to right at an angle of more than 45 degrees up/down) | |
| 4. it doesn't intersect with any of the poles (meaning that, at each pole's position, the function's height must be no smaller than that of the pole) | |
| The cross-sectional area of the tent is the area under this function. Despite | |
| their popularity, the circus troupe doesn't exactly have money to spare on | |
| tent materials (with most of their budget allocated to feeding their flying | |
| elephant star). As such, they'd like to minimize the cross-sectional area of | |
| their tent after placing each of the **N** poles. They'd like you to calculate | |
| the sum of these **N** minimal areas for them. | |
| In this example, three poles are placed one after another. The first is at X = | |
| 20 with height 10. The minimum area of the tent is 100 m2. The second pole is | |
| placed at X = 30 with a height of 15. The minimum area of the tent is now | |
| 268.75 m2. The third pole is at X = 24 with a height of 3. This doesn't change | |
| the minimum area of the tent, which is still 268.75 m2. | |
|   | |
|  | |
| You're given **X1**, and **X2..N** may then be calculated as follows, using | |
| given constants **Ax**, **Bx**, and **Cx** (note that it is guaranteed that | |
| **X1..N** will be distinct): | |
| **Xi** = ((**Ax** * **Xi-1** \+ **Bx**) % **Cx**) + 1 | |
| Similarly, you're given **H1**, and **H2..N** may then be calculated as | |
| follows, using given constants **Ah**, **Bh**, and **Ch**: | |
| **Hi** = ((**Ah** * **Hi-1** \+ **Bh**) % **Ch**) + 1 | |
| ### Input | |
| Input begins with an integer **T**, the number of different tents that the | |
| circus troupe will set up. For each tent, there is first a line containing the | |
| single integer **N**. Then there is a line containing the four space-separated | |
| integers **X1**, **Ax**, **Bx**, and **Cx**. Then there is a line containing | |
| the four space-separated integers **H1**, **Ah**, **Bh**, and **Ch**. | |
| ### Output | |
| For the _i_th tent, print a line containing "Case #**i**: " followed by one | |
| real number. This number is the sum of **N** values, the _j_th of which is the | |
| minimum possible cross-sectional area of the tent after poles 1 through _j_ | |
| have been placed. | |
| Answers that have a relative error of up to 10-6 will be accepted as correct. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 150 | |
| 1 ≤ **N** ≤ 800,000 | |
| 1 ≤ **X1** ≤ 10,000,000 | |
| 0 ≤ **Ax**, **Bx** ≤ 10,000,000 | |
| 1 ≤ **Cx** ≤ 10,000,000 | |
| 1 ≤ **H1** ≤ 100,000 | |
| 0 ≤ **Ah**, **Bh** ≤ 100,000 | |
| 1 ≤ **Ch** ≤ 100,000 | |
| ### Explanation of Sample | |
| In the first case, the cross-sectional areas of the tent after each pole is | |
| erected are 1.0, 1.75, 2.5, 3.25, and 4.0 for a total of 12.50. | |