
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
Team A and Team B are competing in a table tennis match, with each team having three players. The players for Team A are \( A_1, A_2, A_3 \), and the players for Team B are \( B_1, B_2, B_3 \). According to past competition statistics, the win probabilities between players are as follows:

\[
\begin{array}{|c|c|c|}
\hline
Match-up & Probability of Team A Winning & Probability of Team A Losing \\
\hline
A_1 \text{ vs } B_1 & \frac{2}{3} & \frac{1}{3} \\
\hline
A_2 \text{ vs } B_2 & \frac{2}{5} & \frac{3}{5} \\
\hline
A_3 \text{ vs } B_3 & \frac{2}{5} & \frac{3}{5} \\
\hline
\end{array}
\]

Each match yields 1 point for the winning team and 0 points for the losing team. Let \( \xi \) and \( \eta \) be the total scores for Team A and Team B respectively by the end of the competition.

(1) Find the probability distribution of \( \xi \) and \( \eta \).

(2) Determine \( E(\xi) \) and \( E(\eta) \).

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