
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
Let  $n > 1$  be a given integer. An  $n \times n \times n$  cube is composed of  $n^3$  unit cubes. Each unit cube is painted with one colour. For each  $n \times n \times 1$  box consisting of  $n^2$  unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get  $3n$  sets of colours, split into three groups according to the orientation.

It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of  $n$ , the maximal possible number of colours that are present.

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