
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 that the sum of the four numbers on the vertices of a face of a cube is the same for each face, find the total number of different arrangements of the integers $2$ through $9$ on the vertices of the cube, considering arrangements that can be obtained from each other through rotations of the cube as the same.

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