
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 an integer larger than  $1$  and let  $S$  be the set of  $n$ -element subsets of the set  $\{1,2,\ldots,2n\}$ . Determine
\[\max_{A\in S}\left (\min_{x,y\in A, x \neq y} [x,y]\right )\] where  $[x,y]$  is the least common multiple of the integers  $x$ ,  $y$ .

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