
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
Find the largest positive integer  $n$  with the following property: there are rectangles  $A_1, ... , A_n$  and  $B_1,... , B_n,$  on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles  $A_i$  and  $B_i$  are disjoint for all  $i \in \{1,..., n\}$ , but the rectangles  $A_i$  and  $B_j$  have a common point for all  $i, j \in \{1,..., n\}$ ,  $i \ne j$ .

*Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices*

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