| Mr. Fox has just won the lottery! As a result, he's treated himself to some | |
| gifts — a few socks, a few rocks, a few blocks... oh, and the entire Panama | |
| canal system. | |
| The system has **K** canals, the **i**th of which consists of a line of **Ni** | |
| equally-sized sections. The **j**th section of the **i**th canal initially | |
| contains **Wi,j** gallons of water. There's also an initially closed lock (a | |
| retractable wall) between each pair of adjacent sections (that is, between | |
| sections 1 and 2, sections 2 and 3, and so on). As such, there are **Ni**-1 | |
| such locks in the **i**th canal. | |
| The canals are all linked to each other by an additional central hub section | |
| (also of equal size to the other sections), which initially contains **H** | |
| gallons of water. This section is adjacent to the 1st section of each of the | |
| canals, separated by a special initially closed lock. As such, there are **K** | |
| such central locks. | |
| Mr. Fox is relaxing in a yacht (oh, right, he also bought himself one of | |
| those) floating in the central hub section. Just for fun, he'd like to raise | |
| the water level in this section as high as possible. To do so, he may give any | |
| (potentially empty) sequence of commands to his Panama employees, one per | |
| minute. Each command consists of selecting a single lock anywhere in the canal | |
| system and toggling it from being closed to being open (or vice versa). In the | |
| following minute, the water will level out (as water tends to do) by flowing | |
| through open locks such that, for any pair of adjacent sections which are | |
| separated by an open lock, they will end up with equal amounts of water. Mr. | |
| Fox does need to obey the Panama canal system's safety regulations, however, | |
| which impose one restriction on his commands: whenever one of the **K** | |
| central locks adjacent to the central hub section is opened, it must be closed | |
| a minute later and then never re-opened. | |
| Mr. Fox loves watching water flow through his locks, especially when it allows | |
| his yacht to magically rise up. Wheeeee! By commanding his employees | |
| carefully, how much water can Mr. Fox get into the central hub section? | |
| ### Constraints | |
| 1 ≤ **T** ≤ 20 | |
| 1 ≤ **K** ≤ 50 | |
| 0 ≤ **H** ≤ 10^9 | |
| 1 ≤ **Ni** ≤ 100,000 | |
| **Ni** > 1 implies **Ni+1** ≥ 2***Ni** (for 1 ≤ **i** < **K**) | |
| 0 ≤ **Wi,j** ≤ 10^9 | |
| ### Input | |
| Input begins with an integer **T**, the number of canal systems Mr. Fox owns. | |
| For each system, there is first a line containing the space-separated integers | |
| **K** and **H**. Then, **K** lines follow, the **i**th of which contains the | |
| integer **Ni** followed by the space-separated integers **Wi,1** ... | |
| **Wi,Ni**. | |
| ### Output | |
| For the **i**th canal system, print a line containing "Case #**i**: " followed | |
| by the maximum amount of water (in gallons) that can end up in the central hub | |
| section, rounded to 6 decimal places. | |
| Absolute errors of up to 5e-6 will be ignored. | |
| ### Explanation of Sample | |
| In the first case, the optimal solution is to first open and close the lock | |
| between the central hub and canal 1. This leaves the central hub with 0.5 | |
| gallons of water. Then, opening the lock between the central hub and canal 2 | |
| leaves the central hub with 1.25 gallons of water. | |