| Mr. Fox has opened up a fabulous Fock farm! A Fock is a cute little animal | |
| which can have either red, green, or blue fur (these 3 possible colors can be | |
| numbered 1, 2, and 3, respectively). Furthermore, a Fock's fur color can | |
| change every second! | |
| Mr. Fox owns a flock of **N** Focks, with the **i**th one initially having a | |
| color of **Ci**. Every second, if the **i**th Fock currently has a color of | |
| **a**, it will switch to having a color of **b** for the next second with | |
| probability **Pi,a,b**%. All Focks change color simultaneously. | |
| After a very large amount of time has gone by, Mr. Fox will take a single | |
| photo of all of his Focks to help advertise his farm. In particular, he picks | |
| an integer **t** at uniform random from the range [10100, 101000] and waits | |
| that many seconds. He's hoping that the photo will make it look like his farm | |
| has a well-balanced mix of Fock colors — it'll be no good if the photo ends up | |
| featuring a strict majority of a single color (that is, strictly more than | |
| **N**/2 of the Focks having the same color). What's the probability of this | |
| occurring? | |
| ### Constraints | |
| 1 ≤ **T** ≤ 20 | |
| 1 ≤ **N** ≤ 50,000 | |
| 1 ≤ **Ci** ≤ 3 for all **i** | |
| 0 ≤ **Pi,a,b** ≤ 100 for all **i**, **a** and **b** | |
| **Pi,a,1** \+ **Pi,a,2** \+ **Pi,a,3** = 100 for all **i** and **a** | |
| ### Input | |
| Input begins with an integer **T**, the number of Fock farms Mr. Fox has. For | |
| each farm, there is first a line containing the integer **N**. Then, for each | |
| Fock **i**, 4 lines follow. The first of these lines contains the integer | |
| **Ci**. The next three lines contain three space-separated integers each, with | |
| the **b**th integer on the **a**th line being **Pi,a,b**. | |
| ### Output | |
| For the **i**th farm, print a line containing "Case #**i**: " followed by the | |
| probability that the **i**th picture contains a strict majority of some color | |
| of Fock, rounded to 6 decimal places. | |
| Absolute errors of up to 2e-6 will be ignored. | |
| ### Explanation of Sample | |
| In the first case, the first Fock never changes color, so it'll still have | |
| color 1 in the photo. The second Fock is likely to have color 2 for a while, | |
| but by the time the photo is taken, it'll certainly have color 3. The third | |
| Fock will have either color 2 or 3 in the photo, with equal probability. | |
| Therefore, the photo will have a 50% chance of having a strict majority of | |
| color 3, and a 50% chance of no strict majority. | |