
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
Carina is in a tournament in which no game can end in a tie. She continues to play games until she loses 2 games, at which point she is eliminated and plays no more games. The probability of Carina winning the first game is \(\frac{1}{2}\). After she wins a game, the probability of Carina winning the next game is \(\frac{3}{4}\). After she loses a game, the probability of Carina winning the next game is \(\frac{1}{3}\). The probability that Carina wins 3 games before being eliminated from the tournament equals \(\frac{a}{b}\), where the fraction \(\frac{a}{b}\) is in lowest terms. What is the value of \(a+b\)?


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