
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
Let  $n$  be a positive integer and  $A=\{ 1,2,\ldots ,n\}$ . A subset of  $A$  is said to be connected if it consists of one element or several consecutive elements. Determine the maximum  $k$  for which there exist  $k$  distinct subsets of  $A$  such that the intersection of any two of them is connected.

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