
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
Partition a square blackboard into $n^{2}$ small squares with side length 1 using horizontal and vertical lines. Determine the maximum natural number $n$ such that it is always possible to select $n$ small squares so that any rectangle with an area of at least $n$ contains at least one of the selected squares (with the sides of the rectangle being on the grid lines).

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