
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  $S$  be a set with  $n$  elements. Take a positive integer  $k$ . Let  $A_1, A_2, \ldots, A_k$  be any distinct subsets of  $S$ . For each  $i$  take  $B_i = A_i$  or  $B_i = S - A_i$ . Find the smallest  $k$  such that we can always choose  $B_i$  so that  $\bigcup_{i=1}^k B_i = S$ , no matter what the subsets  $A_i$  are.

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