Task:
You will be presented with very difficult competitive programming problems. Your goal is to utilize your full thinking capabilities to solve these problems, providing a detailed, step-by-step explanation of your thought process. Emphasize logical thinking, creativity, and thoroughness in your approach. If you encounter any mistakes along the way, explain them and demonstrate how you correct them.
[Problem Statement
There is a grid with 
H rows and 
W columns. Initially, each integer from 
1 to 
(H×W) is written exactly once in the grid.
Specifically, for 
1≤i≤H and 
1≤j≤W, the cell at the 
i-th row from the top and the 
j-th column from the left contains 
S 
i,j
​
 .
Below, let 
(i,j) denote the cell at the 
i-th row from the top and the 
j-th column from the left.

Determine if it is possible to reach a state where, for all pairs of integers 
(i,j) 
(1≤i≤H,1≤j≤W),
the cell 
(i,j) contains the integer 
((i−1)×W+j) by repeating the following operation at most 
20 times (possibly zero).
If it is possible, print the minimum number of operations required.
If it is impossible in 
20 or fewer operations (including the case where it is impossible no matter how many times the operation is repeated), print 
−1.

Operation: Choose a rectangle of size 
(H−1)×(W−1) from the grid and rotate it 
180 degrees.
More precisely, choose integers 
x and 
y 
(0≤x,y≤1),
and for all pairs of integers 
(i,j) satisfying 
1≤i≤H−1 and 
1≤j≤W−1,
simultaneously replace the integer written in cell 
(i+x,j+y) with the number written in cell 
(H−i+x,W−j+y).

Note that it is only necessary for the integers written in the cells to satisfy the condition; the orientation in which they are written does not matter.

Constraints
3≤H,W≤8
1≤S 
i,j
​
 ≤H×W
If 
(i,j)

=(i 
′
 ,j 
′
 ), then 
S 
i,j
​
 

=S 
i 
′
 ,j 
′
 
​
 
All input values are integers.
Input
The input is given from Standard Input in the following format:

H 
W
S 
1,1
​
  
S 
1,2
​
  
… 
S 
1,W
​
 
S 
2,1
​
  
S 
2,2
​
  
… 
S 
2,W
​
 
⋮
S 
H,1
​
  
S 
H,2
​
  
… 
S 
H,W
​
 
Output
Print the minimum number of operations required to meet the conditions of the problem statement.
If it is impossible to satisfy the conditions with 
20 or fewer operations, output 
−1.

Sample Input 1
Copy
3 3
9 4 3
2 1 8
7 6 5
Sample Output 1
Copy
2
Operating in the following order will satisfy the conditions of the problem statement in 
2 operations.

Operate by choosing the rectangle in the top-left corner. That is, by choosing 
x=0 and 
y=0.
Operate by choosing the rectangle in the bottom-right corner. That is, by choosing 
x=1 and 
y=1.
On the other hand, it is impossible to satisfy the conditions with one or fewer operations, so print 
2.



Sample Input 2
Copy
4 6
15 18 1 14 3 4
23 24 19 8 9 12
13 2 17 6 5 16
21 22 7 20 11 10
Sample Output 2
Copy
-1
It is impossible to satisfy the conditions with 
20 or fewer operations, so print 
−1.

Sample Input 3
Copy
4 6
1 4 13 16 15 18
21 20 9 12 23 10
17 14 5 6 3 2
11 22 7 24 19 8
Sample Output 3
Copy
20
Sample Input 4
Copy
4 3
1 2 3
4 5 6
7 8 9
10 11 12
Sample Output 4
Copy
0]
Instructions:

Carefully Read and Understand the Problem:
Begin by thoroughly reading the problem statement.
Identify all the key components, constraints, and objectives.
Paraphrase the problem in your own words to ensure understanding.
Analyze the Problem:
Break down the problem into smaller parts.
Consider various perspectives and approaches, including innovative or unconventional methods.
Identify any hidden complexities or edge cases that need special attention.
Devise a Plan:
Formulate a step-by-step strategy to solve the problem.
Consider multiple algorithms or data structures that could be applicable.
Evaluate the pros and cons of each potential approach.
Logical Reasoning and Problem-Solving:
Apply logical thinking to progress from one step to the next.
Use diagrams, examples, or analogies if they help illustrate your thought process.
Ensure each step logically follows from the previous one.
Implement the Solution Step-by-Step:
Write pseudocode or code snippets to demonstrate how the solution would be implemented.
Explain each part of the code, detailing how it contributes to solving the problem.
Keep your explanations clear and concise.
Review and Reflect on Your Solution:
After formulating the solution, review each step critically.
Check for any mistakes or oversights.
If you find a mistake, explain what went wrong and how you plan to fix it.
Discuss how correcting the mistake impacts the rest of your solution.
Optimize and Refine:
Analyze the time and space complexity of your solution.
Consider whether there are more efficient algorithms or optimizations.
Explain any improvements and how they enhance your solution.
Provide the Final Answer:
Summarize your approach and the reasoning behind it.
Clearly state the final solution to the problem.
Ensure that all parts of the problem have been addressed.
Demonstrate Deep Understanding:
Reflect on what makes this problem challenging and how your solution effectively tackles these challenges.
Highlight any particularly ingenious or creative aspects of your approach.
Show how logical reasoning led you to the solution.
Remember:

Be Thorough: Leave no step unexplained. Assume the reader is learning from your process.
Be Creative: Don't hesitate to explore unconventional ideas that might lead to a solution.
Be Reflective: Acknowledge and learn from any mistakes, showing how they contribute to your overall understanding.
Utilize Your Full Capabilities: Apply all your knowledge and thinking abilities to solve the problem, demonstrating expertise and depth of thought.