
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
There is a room sized \(13 \times 13\) meters, divided into meter-square grids. It is allowed to place rectangular carpets of arbitrary sizes such that their sides align with the grid lines (including along the room's boundary). Carpets may partially overlap and can even be completely covered by other carpets, but it is not permitted for one carpet to lie entirely on top of or beneath another (even if other carpets are between them). What is the maximum number of carpets that can be placed while adhering to these conditions?

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