
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
A $4 \times 4 \times 4$ cube is cut into 64 smaller $1 \times 1 \times 1$ cubes. Then, 16 of the $1 \times 1 \times 1$ small cubes are painted red. The requirement is that, in any group of 4 small cubes parallel to any edge, exactly 1 cube is painted red. How many different ways are there to paint the cubes? (Painting methods that are the same after rotation are also considered different methods.)

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