
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 $n \geq 2$ be a natural number. For a subset $F$ with $n$ elements of $\{1, \ldots, 2n\}$, define $m(F)$ as the minimum of all the least common multiples (LCM) $\operatorname{lcm}(x, y)$ where $x$ and $y$ are two distinct elements of $F$. Find the maximum value that $m(F)$ can attain.

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