
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 and \( k \) be a positive integer. 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 \( \bigcup B_i = S \).

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