
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 a large cube of dimensions \( 4 \times 4 \times 4 \) composed of 64 unit cubes, select 16 of the unit cubes to be painted red, such that in every \( 1 \times 1 \times 4 \) rectangular prism within the large cube that is composed of 4 unit cubes, exactly 1 unit cube is red. How many different ways are there to select these 16 red unit cubes? Explain why.

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