
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 such that the following hold:
[list=disc]
[*]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
[/list]
(The interior of a polygon does not contain its boundary.)

*Holden Mui*

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