
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 \( A = A_{1} \cup A_{2} \cup \cdots \cup A_{k} \), and let \( S = \{A_{1}, A_{2}, \cdots, A_{k}\} \). If:
(i) The intersection of every \( r \) members of \( S \) is non-empty;
(ii) The intersection of every \( r+1 \) members of \( S \) is empty.

Questions:
(1) What is the minimum possible 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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