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P4 Decomposition - Repovive Standard Round 4 | Repovive | Repovive

11F. P4 Decomposition

3000 pts1s256 MB

You are given a complete graph with $n$ vertices, numbered from $1$ to $n$ .

A $P4$ is a simple path with $4$ distinct vertices and $3$ edges. For example, the path $a-b-c-d$ uses the edges $(a,b)$ , $(b,c)$ , and $(c,d)$ .

Determine whether it is possible to divide all edges of the complete graph into some $P4$ paths, so that every edge belongs to exactly one path.

If it is possible, output one valid division. Using zero paths is allowed.

Constraints

$1 \le t \le 50$
$1 \le n \le 500$

Input

t

\left. \begin{array}{l} n \end{array} \right\} \times t

Output

For each test case, first print Yes if it is possible, or No otherwise.

If it is possible, print an integer $k$ on the next line, the number of paths.

Then print $k$ lines. Each line must contain four distinct integers $a$ , $b$ , $c$ , and $d$ , denoting the path $a-b-c-d$ . If $k=0$ , print no path lines.

Every edge of the complete graph must appear in exactly one printed path.

Samples

Input

Case 1

Case 2

Case 3

Case 4

Case 5

Explanation

Case 1

For $n=1$ , there are no edges, so zero paths are enough.

Case 2

For $n=2$ , the graph has only one edge, so it cannot be divided into paths with three edges.

Case 3

For $n=3$ , it is impossible since a $P4$ needs four distinct vertices.

Case 4

For $n=4$ , the two printed paths use all six edges exactly once.

Case 5

For $n=6$ , the five printed paths use all fifteen edges exactly once.

Output

Case 1

Yes
0

Case 2

No

Case 3

No

Case 4

Case 5

You are given a complete graph with $n$ vertices, numbered from $1$ to $n$ .

A $P4$ is a simple path with $4$ distinct vertices and $3$ edges. For example, the path $a-b-c-d$ uses the edges $(a,b)$ , $(b,c)$ , and $(c,d)$ .

Determine whether it is possible to divide all edges of the complete graph into some $P4$ paths, so that every edge belongs to exactly one path.

If it is possible, output one valid division. Using zero paths is allowed.

$1 \le t \le 50$
$1 \le n \le 500$

t

\left. \begin{array}{l} n \end{array} \right\} \times t

For each test case, first print Yes if it is possible, or No otherwise.

If it is possible, print an integer $k$ on the next line, the number of paths.

Then print $k$ lines. Each line must contain four distinct integers $a$ , $b$ , $c$ , and $d$ , denoting the path $a-b-c-d$ . If $k=0$ , print no path lines.

Every edge of the complete graph must appear in exactly one printed path.

For $n=1$ , there are no edges, so zero paths are enough.

For $n=2$ , the graph has only one edge, so it cannot be divided into paths with three edges.

For $n=3$ , it is impossible since a $P4$ needs four distinct vertices.

For $n=4$ , the two printed paths use all six edges exactly once.

For $n=6$ , the five printed paths use all fifteen edges exactly once.

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11F. P4 Decomposition

Constraints

Input

Output

Samples

11F. P4 Decomposition

Constraints

Input

Output

Samples