
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 natural number \( n \), a \( 3 \times 3 \times 3 \) cube is assembled from 26 unit white cubes and one black cube such that the black cube is situated at the center of the cube. Using \( n^{3} \) such cubes with edge length 3, a larger cube with edge length \( 3n \) is constructed. What is the minimum number of white cubes that need to be painted red so that every white cube shares at least one common vertex with a red cube?

(I. Bogdanov)

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