Problem A
Dhaka Bots

Input: Standard Input

Output: Standard Output

Being the expert programmer that you are, you have decided to put your skills to proper use by making life easier for all vehicle-riding inhabitants of Dhaka city. That is right, you are now in the middle of a project that will eliminate all traffic jams! In a sudden fit of enthusiasm you start to create the control software for those autonomous vehicles, only to realize it is harder than it seemed. You are now to create a small part of that software.

The case you need to handle is simply this: find the time of collision between two vehicles, both going at (possibly different) constant speeds, one of them in a straight line and the other in a circle in counterclockwise direction. You may assume that the vehicles are running on a flat 2D plane, and that they can be modeled as square of a given size with edges parallel to a fixed Cartesian coordinate system at all times.

Input

The input file may contain multiple test cases (at most 100). The first line of the input file consists of a single non-negative integer t, the number of test cases in the file. This line is followed by t test cases, each consisting of exactly two lines of input. The first line describes the vehicle going in a straight line and has the format “s_A x_A y_A v_Ax v_Ay” (without the quotes). s_A is the length of each edge of the square representing this vehicle, (x_A,y_A) is its starting point of the center of the square, (v_Ax, v_Ay) is its velocity in unit distance traveled per unit time. The second line of input in a test case represents the vehicle going round in a circle, and has the format “s_B x_B y_B c_x c_y v_B”. Once again, s_B is the size of the square representing this vehicle, and (x_B,y_B) is the starting point of the center of this square. (c_x, c_y) is the center of the circular path followed by the center of this square, and v_B is its linear speed. All input numbers (other than t) are real numbers with at most two decimal places. They will also satisfy the following constraints: |x_A|, |y_A|, |x_B|, |y_B|, |c_x| |c_y| ≤ 20000, |v_Ax|, |v_Ay| ≤ 200, 0 ≤ v_B ≤ 200, 0 < s_A, s_B ≤ 100. It is also guaranteed that the two squares do not already overlap in their given starting states.

Output

There should be a single line of output for each test case in the input file, and it should have the format “Case c: t”. c is the case number, starting from 1, and t is the time it takes before the vehicles collide. Values within an absolute error of 10^-5 of the actual answer will be accepted. If they never collide after starting in the state given, t should be replaced with the string “never”. Note that, for this problem, just grazing across the sides or just touching corners is not considered a collision.

Sample Input Output for Sample Input

2 2 -10 -10 1 1 2 0 5 0 0 10 2 -10 -10 1 1 2 0 5 0 0 1

Case 1: 7.216133 Case 2: never

Problemsetter: Samee Zahur

Special Thanks to: Manzurur Rahman Khan