
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  $A$  be a family of subsets of  $\{1,2,\ldots,n\}$  such that no member of  $A$  is contained in another. Sperner’s Theorem states that  $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$ . Find all the families for which  the equality holds.

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