
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 colored with one color. For each \( n \times n \times 1 \) rectangular prism (in each of the three possible orientations), consider the set of colors appearing in this rectangular prism (each color appears only once), resulting in \( 3n \) color sets, which are divided into three groups based on orientation. For any color set in one 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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