
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 \(\mathcal{A}\) be a set of subsets of \(\llbracket 1, 2n \rrbracket\) such that for all distinct \(B\) and \(C\) in \(\mathcal{A}\), \(B\) is not included in \(C\). (In other words, \(\mathcal{A}\) is an antichain.)

Show the following inequality:

$$
\sum_{B \in \mathcal{A}} \frac{1}{\binom{n_{B}}{n}} \leq 1
$$

Deduce that \(|\mathcal{A}| \leq \binom{n}{\left\lfloor \frac{n}{2} \right\rfloor}\).

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