
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 \( B_{k}(n) \) be the largest possible number of elements in a 2-separable \( k \)-configuration of a set with \( 2n \) elements \((2 \leq k \leq n)\). Find a closed-form expression (i.e., an expression not involving any sums or products with a variable number of terms) for \( B_{k}(n) \).

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