
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
Ana, Beto e Carolina are going to participate in the television program "Guess the Color of Your Hat". In the program, they position themselves in a circle, and each will have a blue or green hat placed on their head. Each can see the hats of the others, but not the color of their own hat. Then, each writes one of three options on a paper: "blue", "green", or "pass". If all who wrote "blue" or "green" correctly guess the color of their hat, they win a new car. If any one of them guesses the hat color and is wrong, they all lose. If all of them write "pass", they also lose. They cannot communicate or see each other's papers. 

Carolina suggests: "None of us should write 'pass'; we should guess between 'blue' and 'green' because if we all pass, we lose." 

Beto replies: "I disagree, it is better if only Ana guesses her hat color while Carolina and I write 'pass'. This way, our chance of winning will be higher." 

Ana proposes: "I have an idea. If we use my strategy, our probability of winning will be $\frac{3}{4}$. 

(a) According to Carolina's idea, what is the probability of winning the car?

(b) According to Beto's idea, what is the probability of winning the car?

(c) Provide an example of a possible strategy of Ana that makes the probability of winning the car $\frac{3}{4}$.

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