
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 the set
\[
S = \{ (x, y, z) : x, y, z \in \{0, 1, \cdots, n\}, x + y + z > 0 \}
\]
This is a set of points in three-dimensional space with \((n+1)^3 - 1\) elements. What is the minimum number of planes required such that their union covers \( S \) but does not contain the point \((0,0,0)\)?

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