
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 that \( n \) is a positive integer,
$$
\begin{aligned}
S= & \{(x, y, z) \mid x, y, z \in \{0,1, \cdots, n\}, \\
& x+y+z>0\}
\end{aligned}
$$
which is a set of \((n+1)^3-1\) points in three-dimensional space, find the minimum number of planes needed to cover the set \( S \) but not include the point \( (0,0,0) \).

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