
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 \( n \geq 2 \). An \( n \times n \times n \) cube is divided into \( n^{3} \) unit cubes. Each unit cube is painted in one color. For each \( n \times n \times 1 \) rectangular prism (in each of the three possible directions), consider the set of colors that appear in this prism (each color appears only once), yielding \( 3n \) color sets, partitioned into three groups based on direction. For each color set in any group, it also appears in each of the other two groups. Find the maximum number of colors (expressed in terms of \( n \)).

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