
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
In a different scenario, each vertex of a cube needs to be labeled with a unique integer from $2$ through $9$, ensuring that the sum of the four numbers on each face is constant across all faces. Similar to the original problem, arrangements that can be achieved through rotations of the cube are considered the same. Determine how many distinct labeling arrangements meet these criteria.
- **A**: 1
- **B**: 2
- **C**: 3
- **D**: 6
- **E**: 12

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