
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
In 2011, the International Mathematical Union officially announced that March 14th of each year would be celebrated as the International Day of Mathematics, in honor of the ancient Chinese mathematician Zu Chongzhi's approximation of pi. To celebrate this occasion, a school organized a Math Carnival, which included a prize-winning challenge game. Contestants would attempt challenges in the order of the first, second, and third rounds. Successful completion of each round would award 5, 10, and 20 "learning beans" respectively. The game also allowed contestants to either take their earned beans and end the game after completing any round or to proceed to the next round. If a contestant failed at any round, they would lose all their beans, and the game would end. Assume the probabilities of a contestant successfully completing the first, second, and third rounds are $\frac{3}{4}$, $\frac{2}{3}$, and $\frac{1}{2}$ respectively, and the probability of choosing to proceed to the next round is $\frac{1}{2}$ for each round. The success in each round is independent of the others.  
(1) Calculate the probability of a contestant winning 5 learning beans;  
(2) Calculate the probability of contestant A successfully completing the first round but ending up with zero learning beans.

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