#P1521E. Nastia and a Beautiful Matrix
Nastia and a Beautiful Matrix
Description
You like numbers, don't you? Nastia has a lot of numbers and she wants to share them with you! Isn't it amazing?
Let $a_i$ be how many numbers $i$ ($1 \le i \le k$) you have.
An $n \times n$ matrix is called beautiful if it contains all the numbers you have, and for each $2 \times 2$ submatrix of the original matrix is satisfied:
- The number of occupied cells doesn't exceed $3$;
- The numbers on each diagonal are distinct.
Make a beautiful matrix of minimum size.
The first line contains a single integer $t$ ($1 \le t \le 10\,000$) — the number of test cases.
The first line of each test case contains $2$ integers $m$ and $k$ ($1 \le m, k \le 10^5$) — how many numbers Nastia gave you and the length of the array $a$, respectively.
The second line of each test case contains $k$ integers $a_1, a_2, \ldots, a_{k}$ ($0 \le a_i \le m$, $a_1 + a_2 + \ldots + a_{k} = m$), where $a_i$ is how many numbers $i$ you have.
It's guaranteed that the sum of $m$ and $k$ in one test doesn't exceed $2 \cdot 10^5$.
For each $t$ test case print a single integer $n$ — the size of the beautiful matrix.
In the next $n$ lines print $n$ integers $b_{i, j}$ ($0 \le b_{i, j} \le k$; if position is empty, print $b_{i, j} = 0$) — the beautiful matrix $b$ you made up.
Input
The first line contains a single integer $t$ ($1 \le t \le 10\,000$) — the number of test cases.
The first line of each test case contains $2$ integers $m$ and $k$ ($1 \le m, k \le 10^5$) — how many numbers Nastia gave you and the length of the array $a$, respectively.
The second line of each test case contains $k$ integers $a_1, a_2, \ldots, a_{k}$ ($0 \le a_i \le m$, $a_1 + a_2 + \ldots + a_{k} = m$), where $a_i$ is how many numbers $i$ you have.
It's guaranteed that the sum of $m$ and $k$ in one test doesn't exceed $2 \cdot 10^5$.
Output
For each $t$ test case print a single integer $n$ — the size of the beautiful matrix.
In the next $n$ lines print $n$ integers $b_{i, j}$ ($0 \le b_{i, j} \le k$; if position is empty, print $b_{i, j} = 0$) — the beautiful matrix $b$ you made up.
Samples
2
3 4
2 0 0 1
15 4
2 4 8 1
2
4 1
0 1
5
3 0 0 2 2
3 2 3 3 0
0 1 0 4 0
3 0 0 0 0
2 1 3 3 3
Note
Note that $0$ in this problem represents a blank, not a number.
Examples of possible answers for the first test case:
$\begin{array}{cc} 1 & 1 \\ 4 & 0 \\ \end{array} \hspace{0,5cm} \begin{array}{cc} 1 & 4 \\ 1 & 0 \\ \end{array} \hspace{0,5cm} \begin{array}{cc} 4 & 0 \\ 1 & 1 \\ \end{array}$
Examples of not beautiful matrices for the first test case:
$\begin{array}{cc} 1 & 0 \\ 4 & 1 \\ \end{array} \hspace{0,5cm} \begin{array}{cc} 4 & 1 \\ 7 & 1 \\ \end{array} \hspace{0,5cm} \begin{array}{cc} 1 & 0 \\ 4 & 0 \\ \end{array}$
The example of the not beautiful matrix for the second test case:
$\begin{array}{cc} 3 & 4 & 0 & 2 & 2 \\ 3 & 2 & 3 & 3 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 \\ 2 & 1 & 3 & 3 & 3 \\ \end{array}$
Everything is okay, except the left-top submatrix contains $4$ numbers.