
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 the picture, there is a structure made up of twelve identical cubes. How many different positions can we move the dark cube to, so that the surface area of the assembled body remains unchanged?

Like in the original structure, in the new structure, the cubes must touch each other with their entire faces. The positions of the light cubes cannot be changed.

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