
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
Two players, A and B, are competing in a game. Player A is the younger player, and the rules are as follows: In odd-numbered rounds, the probability of A winning is $\frac{2}{3}$; in even-numbered rounds, the probability of B winning is $\frac{2}{3}$. There are no draws in any round. The game ends when one player has won 2 more rounds than the other, with a maximum of 8 rounds. If the game reaches the 8th round without a clear winner, A is declared the winner.

(I) Calculate the probability that the 4th round determines the winner and A wins;
(II) Calculate the mathematical expectation of the total number of rounds played, denoted as $\xi$.

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