
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
Aline received a cube made of $4 \times 4 \times 4$ small white cubes as a gift.

Without separating the small cubes, Aline decided to paint all the faces of her cube with red paint.

a) Determine how many small cubes have exactly one of their faces painted red.

b) Determine how many small cubes have exactly two of their faces painted red.

Sometime later, Aline asked her father for an even larger cube to paint in the same way she did with the previous cube.

c) After completing the painting, Aline separated the small cubes. She noticed that the number of small cubes with no faces painted red was three times the number of small cubes with two faces painted red. Find the size of the new cube that Aline received.

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