You're about to put on an exciting show at your local circus — a parkour demonstration! N platforms with adjustable heights have been set up in a row, and are numbered from 1 to N in order from left to right. The initial height of platform i is Hi metres.
When the show starts, M parkourists will take the stage. The _i_th parkourist will start at platform Ai, with the goal of reaching a different platform Bi. If Bi > Ai, they'll repeatedly jump to the next platform to their right until they reach Bi. If Bi < Ai, they'll instead repeatedly jump to the next platform to their left until they reach Bi. All of the parkourists will complete their routes simultaneously (but don't worry, they've been trained well to not impede one another).
Not all parkourists are equally talented, and there are limits on how far up or down they can jump between successive platforms. The _i_th parkourist's maximum upwards and downwards jump heights are Ui and Di, respectively. This means that they're only able to move directly from platform x to some adjacent platform y if Hx - Di ≤ Hy ≤ Hx + Ui, where Hx and Hy are the current heights of platforms x and y, respectively.
With the show about to begin, a disastrous flaw has just occurred to you — it
may not be possible for all of the parkourists to actually complete their
routes with the existing arrangement of platforms! If so, you will need to
quickly adjust some of the platforms' heights first. The height of each
platform may be adjusted upwards or downwards at a rate of 1 metre per second,
to any non-negative real-valued height of your choice, and multiple platforms
may be adjusted simultaneously. As such, if the initial height of platform i
is Hi and its final height is Pi, then the total time required to make
your chosen height adjustments will be max{|Hi - Pi|} over
i=1..N.
Determine the minimum amount of time required to set up the platforms such that all M parkourists will then be able to complete their required routes. Note that you may not perform further height adjustments once the show starts. The platform heights must all remain constant while all M parkourists complete their routes.
In order to reduce the size of the input data, you're given H1 and H2. H3..N may then be generated as follows using given constants W, X, Y, and Z (please watch out for integer overflow during this process):
Hi = (W * Hi-2 + X * Hi-1 + Y) % Z (for i=3..N)
Input
Input begins with an integer T, the number of shows. For each show, there is first a line containing the space-separated integers N and M. The next line contains the space-separated integers H1, H2, W, X, Y, and Z. Then, M lines follow. The _i_th of these lines contains the space-separated integers Ai, Bi, Ui, and Di.
Output
For the _i_th show, print a line containing "Case #i: " followed by 1 real number, the minimum amount of time required to set up the platforms (in seconds). Absolute and relative errors of up to 10-6 will be ignored.
Constraints
1 ≤ T ≤ 85
2 ≤ N ≤ 200,000
1 ≤ M ≤ 20
0 ≤ Hi < Z
0 ≤ W, X, Y < Z
1 ≤ Z ≤ 1,000,000
1 ≤ Ai, Bi ≤ N
0 ≤ Ui, Di ≤ 1,000,000
Ai, ≠ Bi
Explanation of Sample
In the first case, H = [0, 10]. You can increase the first platform's height by 3.5 and decrease the second's by 3.5 in 3.5 seconds, yielding P = [3.5, 6.5]. The single parkourist will then be able to successfully complete their route from platform 1 to platform 2 by jumping upwards by a height of at most 3.
In the second case, H = [50, 59, 55, 51, 47]. One optimal possibility is P = [54.0, 54.5, 53.5, 52.5, 51.5].
In the third case, H = [46, 38, 38, 22, 8].
In the fourth case, H = [53, 25, 24, 81, 77, 40, 29, 21].