
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 positive integer \( n \), using 26 white unit cubes and 1 black unit cube, form a \( 3 \times 3 \times 3 \) cube. Then, form a cube with side length \( 3n \) using \( n^3 \) such \( 3 \times 3 \times 3 \) cubes. The question is: What is the minimum number of white unit cubes that need to be painted red so that every remaining white unit cube has at least one common point with at least one red unit cube?

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