
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 wooden cube, $n$ units on each side, is painted red on four of its six faces (the top and bottom faces are left unpainted), and then cut into $n^3$ smaller unit cubes. Exactly one-third of the total number of faces of these unit cubes are red. What is the value of $n$, and what is the volume of the original cube in cubic units?

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