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 ith one initially having a color of Ci. Every second, if the ith 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 bth integer on the ath line being Pi,a,b.
Output
For the ith farm, print a line containing "Case #i: " followed by the probability that the ith 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.