
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]


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Let \( n \) be a positive integer. A set of points \( S \) in three-dimensional space satisfies the following properties:

1. There do not exist \( n \) planes such that every point in the set \( S \) lies on at least one of these \( n \) planes.
2. For every point \( X \in S \), there exist \( n \) planes such that every point in \( S \setminus \{X\} \) lies on at least one of these \( n \) planes.

Find the minimum and maximum number of points in the set \( S \).

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