
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]


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To address the threat of AIDS to humanity, now three research institutes, designated as Institute A, Institute B, and Institute C, are independently developing an AIDS vaccine. The probabilities of successfully developing a vaccine are respectively $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$. Calculate:  
(1) The probability that exactly one research institute is successful in developing the vaccine;  
(2) In order to achieve at least a $\frac{99}{100}$ probability that the vaccine is successfully developed (meaning at least one research institute is successful), how many institutes similar to Institute B are minimally required? (Reference data: $\lg 2=0.3010$, $\lg 3=0.4771$)

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