
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 an  $n$ -element set  $S$ , several subsets  $A_1, A_2, \ldots , A_k$  are distinguished, each consists of at least two, but not all elements of  $S$ . What is the largest  $k$  that it’s possible to write down the elements of  $S$  in a row in the order such that we don’t find all of the element of an  $A_i$  set in the consecutive elements of the row? 

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