
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
If the family of subsets $\triangle$ of an $n$-element set $X = \{1, 2, \cdots, n\}$ is an $S$ family (Sperner family), then the maximum size of the family $\mathcal{A}$ is at most $C_n^{\left\lfloor \frac{n}{2} \right\rfloor}$, that is,
$$
\max_{\text{family}} |\mathscr{A}| = C_n^{\left\lfloor \frac{n}{2} \right\rfloor}.
$$

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