
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 $A_{1}, A_{2}, \cdots, A_{k}$ be a family of subsets of $X=\{1,2, \cdots, n\}$. The family of subsets is called separating if for any pair of elements $i, j$ in $X$, there is a subset in $A_{1}, A_{2}, \cdots, A_{k}$ that contains exactly one of $i$ or $j$. Find the minimum $k$ such that there exists a family of subsets $A_{1}, A_{2}, \cdots, A_{k}$ that is both covering and separating.

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