23447번 - Mr. Panda and Blocks 스페셜 저지다국어

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.