
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
Players A, B, C, and D compete in a badminton singles match. The competition adopts a round-robin format, where each pair of players plays a match. Each match is played in a best-of-three games format, meaning the player who wins two games first wins the match, and the match ends immediately. It is assumed that the results of each match and each game do not affect each other.  
$(1)$ If player A has a winning probability of $\frac{2}{3}$ per game against player B, find the probability of player A winning the match against player B.  
$(2)$ If player A has winning probabilities of $\frac{1}{2}$, $\frac{2}{3}$, and $\frac{3}{4}$ against players B, C, and D respectively, determine the opponent for player A in the second match to maximize the probability of player A winning exactly two consecutive matches out of three.

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