
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 a table tennis match between players A and B, they follow the rules that: the winner of each game gets 1 point and the loser gets 0 points. The match stops when one player has 2 more points than the other or after a maximum of 6 games. Let the probability that A wins each game be $\frac{3}{4}$ and the probability that B wins each game be $\frac{1}{4}$, and assume the outcomes of each game are independent. Find the expected number of games $\xi$ played when the match stops, denoted as $\mathrm{E} \xi$.

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