
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
Consider a set  $X$  with  $|X| = n\geq 1$  elements. A family  $\mathcal{F}$  of distinct subsets of  $X$  is said to have property  $\mathcal{P}$  if there exist  $A,B \in \mathcal{F}$  so that  $A\subset B$  and  $|B\setminus A| = 1$ .

i) Determine the least value  $m$ , so that any family  $\mathcal{F}$  with  $|\mathcal{F}| > m$  has property  $\mathcal{P}$ .
ii) Describe all families  $\mathcal{F}$  with  $|\mathcal{F}| = m$ , and not having property  $\mathcal{P}$ .

(*Dan Schwarz*)

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