
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
A and B play a game with the following rules: In the odd-numbered rounds, A has a winning probability of $\frac{3}{4}$, and in the even-numbered rounds, B has a winning probability of $\frac{3}{4}$. There are no ties in any round, and the game ends when one person has won 2 more rounds than the other. What is the expected number of rounds played until the game ends?

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