
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
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a *box*. Two boxes *intersect* if they have a common point in their interior or on their boundary. Find the largest  $ n$  for which there exist  $ n$  boxes  $ B_1$ ,  $ \ldots$ ,  $ B_n$  such that  $ B_i$  and  $ B_j$  intersect if and only if  $ i\not\equiv j\pm 1\pmod n$ .

*Proposed by Gerhard Woeginger, Netherlands*

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