
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$  be a positive integer. Consider
\[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \}
\]
as a set of  $ (n \plus{} 1)^{3} \minus{} 1$  points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains  $ S$  but does not include  $ (0,0,0)$ .

*Author: Gerhard Wöginger, Netherlands*

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