
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 and B are playing a game of chess, with the rule that the first to win two games more than the other player wins the match, and the match ends immediately. If, after six games, no winner is determined, the winner of the first game is declared the overall winner and the match ends. During the match, the probability of player A winning any game is $\frac{2}{3}$, while the probability of player B winning is $\frac{1}{3}$, and the outcome of each game is independent. Determine:  
(1) The probability that the match ends after two games with player A's victory;  
(2) The probability that the match ends exactly after four games;  
(3) The probability that player A wins the match overall.

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