
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
The central square of the City of Mathematicians is an  $n\times n$  rectangular shape, each paved with  $1\times 1$  tiles. In order to illuminate the square, night lamps are placed at the corners of the tiles (including the edges of the rectangle) in such a way that each night lamp illuminates all the tiles in its corner.
Determine the minimum number of night lamps such that even if one of those night lamps does not work, it is possible to illuminate the entire central square with them.


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