
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 $n \in \mathbb{N}$. We define

$$
S=\left\{(x, y, z) \in\{0, \ldots, n\}^{3} \mid x+y+z>0\right\}
$$

a set of $(n+1)^{3}-1$ points in three-dimensional space. Determine the minimal number of planes whose union contains $S$ but not the point $(0,0,0)$.

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