
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Given a chessboard with numbers as shown in the image below:

$\left(\begin{array}{llllllll}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0\end{array}\right)$

How many arrangements of 8 rooks exist such that the rooks do not attack each other and the squares they occupy contain all the numbers from 0 to 7?

Present the answer in LaTex format: \boxed{Your answer}
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