| The princess of the kingdom of Hackadia has been kidnapped by an evil dragon. | |
| As always the prince Z.A.Y. is going to try to rescue her. The evil dragon is | |
| keeping the princess prisoner in his deepest dungeon, and the prince has to | |
| solve a puzzle to get her out safely. | |
| There are two straight boards in front of the dungeon, both divided into a | |
| large number of sections. Each section contains a sparkling gemstone. Each | |
| stone has a color. We will denote these colors with numbers. | |
| Let's say these boards **Board1** and **Board2** contain **N** and **M** | |
| sections respectively. Let's call a pair of integers **x**, **y** a magic pair | |
| if they have the following properties: | |
| 1. 0 ≤ **x** < **N** | |
| 2. 0 ≤ **y** < **M** | |
| 3. The set of different colors in **Board1**[0...**x**] equals the set of different colors in **Board2**[0...**y**] | |
| The prince has asked you to find out how many magic pairs exist for the given | |
| two boards, so he can free the princess and become the hero. He will take all | |
| the glory from this, so you will have to make do with points in this | |
| competition as payment for your help. | |
| Since the numbers **N** and **M** might be very large, the colors of the | |
| gemstones will be supplied through a pseudo random generator. This works as | |
| follows: | |
| **Board1**[0] = **x1** | |
| **Board2**[0] = **x2** | |
| **Board1**[**i**] = (**a1** * **Board1**[(**i**-1) % **N**] + **b1** * **Board2**[(**i**-1) % **M**] + **c1**) % **r1**, for 0 < **i** < **N** | |
| **Board2**[**i**] = (**a2** * **Board1**[(**i**-1) % **N**] + **b2** * **Board2**[(**i**-1) % **M**] + **c2**) % **r2**, for 0 < **i** < **M** | |
| ### Input | |
| The first line of the input consists of a single integer **T**, the number of | |
| test cases. | |
| Each test case starts with a line containing the integers **N**, **M** | |
| The second line of each test case contains five integers **x1**, **a1**, | |
| **b1**, **c1**, **r1** | |
| The third line of each test case contains five integers **x2**, **a2**, | |
| **b2**, **c2**, **r2** | |
| ### Output | |
| For each test case **i** numbered from 1 to **T**, output "Case #**i**: ", | |
| followed by the number of magic pairs for the two boards. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 20 | |
| 1 ≤ **N**, **M** ≤ 10^6 | |
| 0 ≤ **x1**, **x2**, **a1**, **a2**, **b1**, **b2**, **c1**, **c2** ≤ 10^9 | |
| 1 ≤ **r1**, **r2** ≤ 10^9 | |
| ### Examples | |
| The first example produces the following boards: | |
| **Board1** = [0, 3, 2, 0, 4, 2, 1, 3] | |
| **Board2** = [0, 4, 2, 1, 4, 3] | |
| There are 3 magic pairs: | |
| pair(0, 0) ==> (0) | |
| pair(6, 5) ==> (0, 1, 2, 3, 4) | |
| pair(7, 5) ==> (0, 1, 2, 3, 4) | |