
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 have a table tennis team match. Each team has three players, and each player plays once. Team A's three players are \( A_{1}, A_{2}, A_{3} \) and Team B's three players are \( B_{1}, B_{2}, B_{3} \). The winning probability of \( A_{i} \) against \( B_{j} \) is \( \frac{i}{i+j} \) for \( 1 \leq i, j \leq 3 \). The winner gets 1 point. What is the maximum possible expected score for Team A?

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