
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
In set operations, a set is in one-to-one correspondence with its complement in the universal set, forming a "set pair", which is often used in solving set problems. Now, suppose the universal set $U$ contains $11$ elements. For $k$ different subsets $A_{1}$, $A_{2}$, $\ldots$, $A_{k}$ of set $U$, where the intersection of any two subsets is not empty, and any other subset of $U$ has an empty intersection with at least one of $A_{1}$, $A_{2}$, $\ldots$, $A_{k}$, then $k=\_\_\_\_\_\_$.

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