
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 \( i \not \equiv j \pm 1 \pmod{n} \).

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