
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 \( A = A_{1} \cup A_{2} \cup \cdots \cup A_{k} \), \( S = \{A_{1}, A_{2}, \cdots, A_{k}\} \). If (i) the intersection of any \( r \) elements in \( S \) is non-empty; (ii) the intersection of any \( r+1 \) elements in \( S \) is empty, then:

(1) What is the minimum value of \( |A| \)?

(2) When \( |A| \) is minimized, what is the value of \( |A_{i}| \)?

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