
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 following set of \((n+1)^{3}-1\) points in three-dimensional space:

\[ 
S=\{(x, y, z): x, y, z \in \{0,1, \ldots, n\}, x+y+z>0\} 
\]

Determine the minimum number of planes whose union contains all points of the set \( S \), but does not contain the point \((0,0,0)\).

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