406525: GYM102431 I Mr. Panda and Blocks

Mr. Panda recently received a bucket of toy blocks as his birthday gift. Each block is a $$$1 \times 1\times 2$$$ cuboid, which is constructed by a pair of face-to-face $$$1 \times 1 \times 1$$$ colored cubes. There are $$$n$$$ types of colors, labeled as $$$1, 2, \ldots, n$$$.

Mr. Panda checked all of the blocks, and he found that he had just $$$\frac{n \times (n+1)}{2}$$$ blocks and each of these blocks was painted with a unique pair of colors. That is, for each pair of colors $$$(i, j)$$$ $$$(1 \leq i \leq j \leq n)$$$, he had exactly one block with one cube colored $$$i$$$, and the other colored $$$j$$$.

Mr. Panda plans to build a fantastic castle with these blocks today.

Firstly, he defines an attribute called connected:

1. If cube A shares a face with cube B, they are connected.
2. If cube A and cube B are connected, and cube B and cube C are connected, then cube A and cube C are connected.
3. If all pairs of cubes in the castle are connected, the castle is connected.

Then he comes up with the following requirements:

1. The whole castle should be connected.
2. For any color $$$i$$$, if only consider the cubes with that color, the sub-castle formed by these cubes should also be connected.

However, after many attempts, Mr. Panda still cannot build such a castle. So he turns to you for help. Could you please help Mr. Panda to build a castle which meets all his requirements?

The first line of the input gives the number of test cases, $$$T$$$ ($$$1 \leq T \leq 10$$$). $$$T$$$ test cases follow.

For each test case, one line contains an integer $$$n$$$ ($$$1 \leq n \leq 200$$$), representing the number of colors.

For each test case, first output one line containing "Case #x:", where x is the test case number (starting from 1).

If it's impossible to build a castle that satisfies Mr. Panda's requirements, output a single line containing "NO" (quotes for clarity).

If it's possible to build the castle, first output a single line containing "YES" (quotes for clarity).

Then, output $$$\frac{n \times (n+1)}{2}$$$ lines describing the coordinates of all the blocks. Each of these lines should be outputted in the form of $$$i,j,x_i,y_i,z_i,x_j,y_j,z_j$$$ $$$(1 \leq i \leq j \leq n, 0 \leq x_i, y_i, z_i, x_j, y_j, z_j \leq 10^{9}$$$), which means for the block $$$(i,j)$$$, the cube with color $$$i$$$ is located at $$$(x_i,y_i,z_i)$$$ and the other cube with color $$$j$$$ is located at $$$(x_j,y_j,z_j)$$$. You should make sure that each pair of $$$(i, j)$$$ occurs exactly once in your answer.

In case there is more than one solution, any of them will be accepted.

2
3
4

Case #1:
YES
1 1 1 1 0 1 2 0
1 2 1 3 0 1 4 0
1 3 2 1 0 3 1 0
2 2 2 2 0 2 3 0
2 3 2 4 0 3 4 0
3 3 3 2 0 3 3 0
Case #2:
YES
1 2 1 3 0 1 4 0
1 1 1 2 0 1 1 0
1 3 2 1 0 2 2 0
2 3 2 4 0 2 3 0
1 4 3 1 0 4 1 0
3 3 3 2 0 3 3 0
2 2 3 4 0 3 4 1
4 4 4 2 0 5 2 0
3 4 4 3 0 5 3 0
2 4 4 4 0 5 4 0