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 $1, 2, \\dots, n$.
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 $(i, j)$ $(1 <= i <= j <= n)$, 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_:
Then he comes up with the following requirements:
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, $T$ ($1 <= T <= 10$). $T$ test cases follow.
For each test case, one line contains an integer $n$ ($1 <= n <= 200$), 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 $i, j, x_i, y_i, z_i, x_j, y_j, z_j$ $(1 <= i <= j <= n, 0 <= x_i, y_i, z_i, x_j, y_j, z_j <= 10^9$), 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.
## Input
The first line of the input gives the number of test cases, $T$ ($1 <= T <= 10$). $T$ test cases follow.For each test case, one line contains an integer $n$ ($1 <= n <= 200$), representing the number of colors.
## Output
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 $i, j, x_i, y_i, z_i, x_j, y_j, z_j$ $(1 <= i <= j <= n, 0 <= x_i, y_i, z_i, x_j, y_j, z_j <= 10^9$), 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.
[samples]
**Definitions**
Let $ n \in \mathbb{Z} $ be the number of colors.
Let $ \mathcal{B} = \{ (i, j) \mid 1 \leq i < j \leq n \} $ be the set of block types, where each block $(i,j)$ connects two cubes of colors $i$ and $j$.
There are $ |\mathcal{B}| = \frac{n(n+1)}{2} $ blocks.
**Constraints**
1. $ 1 \leq n \leq 200 $
2. Each block $(i,j) \in \mathcal{B}$ must be represented by two unit cubes at integer coordinates in $ \mathbb{Z}^3 $, with one cube colored $i$, the other $j$, and the two cubes are face-adjacent (i.e., Manhattan distance 1).
3. The entire structure must be connected (i.e., the graph where vertices are cubes and edges are face-adjacencies is connected).
**Objective**
For each test case, either:
- Output "NO" if no such connected arrangement exists, or
- Output "YES" followed by $ \frac{n(n+1)}{2} $ lines, each specifying the 3D coordinates of the two cubes of block $(i,j)$, such that:
- All blocks are used exactly once,
- Each block’s two cubes are face-adjacent,
- The entire assembly is connected.