
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
On a  $3 \times 3$  board the numbers from  $1$  to  $9$  are written in some order and without repeating. We say that the arrangement obtained is *Isthmian* if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.

Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.

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