Problem Description

Given a $2 \times 2$ numeric matrix $\left[\begin{matrix}a& b\\ c&d\end{matrix}\right]$, we define its second-order determinant value as $a \times d - c \times b$.

For a numeric matrix $A$ with $n$ rows and $m$ columns, let $A_{i,j}$ denote the number in row $i$ and column $j$ ($1 \leq i \leq n$, $1 \leq j \leq m$). We define its second-order determinant matrix as an $(n-1)$-row, $(m-1)$-column matrix $B$, where $B_{i,j}$ is the second-order determinant of the matrix $\left[\begin{matrix}A_{i,j}&A_{i,j+1}\\ A_{i+1,j}&A_{i+1,j+1}\end{matrix}\right]$, that is, $B_{i,j} = A_{i,j} \times A_{i+1,j+1} - A_{i,j+1}\times A_{i+1,j}$.

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Now, given matrix $A$. Taking the second-order determinant matrix of $A$ gives matrix $B$, taking the second-order determinant matrix of $B$ gives matrix $C$, and taking the second-order determinant matrix of $C$ gives matrix $D$.

The problem will input the name of the matrix to compute, which is one of the four letters $\texttt{A,B,C,D}$. Please output the corresponding matrix.

Input Format

The first line contains two integers, representing the number of rows and columns of matrix $A$, $n, m$.
Then follow $n$ lines, each with $m$ integers. The number in row $i$ and column $j$ represents $A_{i,j}$.
The last line contains a character, representing the requested matrix name $q$.

Output Format

First output two integers $r, c$, representing the number of rows and columns of the requested matrix.
Then output $r$ lines, each with $c$ integers. The number in row $i$ and column $j$ represents the value at row $i$ and column $j$ of the requested matrix.

4 4
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
A

4 4
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16

4 4
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
B

3 3
-4 -4 -4
-4 -4 -4
-4 -4 -4

4 4
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
C

2 2
0 0
0 0

Hint

Explanation for Sample 1

Matrix $A$ is exactly the input matrix.

Explanation for Sample 2

$B_{1,1}=A_{1,1}\times A_{2,2}-A_{1,2}\times A_{2,1}=1\times6-2\times5=-4$.

Similarly, you can compute that the values at other positions are also $-4$.

Explanation for Sample 3

$C_{1,1} = B_{1,1} \times B_{2,2} - B_{1,2}\times B_{2,1} = (-4)\times(-4) - (-4)\times(-4) = 0$.

Similarly, you can compute that the values at other positions are also $0$.

Constraints

For all testdata, it is guaranteed that $4 \leq n, m \leq 10^3$, and $1 \leq A_{i,j} \leq 100$.

Translated by ChatGPT 5