
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 table tennis match. The winner of each game scores 1 point, and the loser scores 0 points. The match ends when one player has 2 points more than the other or after 6 games, whichever comes first. The probability of A winning a game is $\frac{2}{3}$, and the probability of B winning a game is $\frac{1}{3}$. The outcomes of the games are independent. Find the expected number of games $\xi$ played when the match ends, $\mathrm{E} \xi$.

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