
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. Find the smallest positive integer \( k \) such that for any set \( S \) of \( n \) points in the interior of the unit square, there exists a set of \( k \) rectangles satisfying:
- The sides of each rectangle are parallel to the sides of the unit square.
- Each point in \( S \) is not in the interior of any rectangle.
- Each point in the interior of the unit square but not in \( S \) is in the interior of at least one of the \( k \) rectangles.
(The interior of a polygon does not contain its boundary.)

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