
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 playing a game of Go. They agree that the first player to win two consecutive games wins the match. If after playing 5 games there is no consecutive win, the player with more wins is declared the winner. Assume that the probability of player A winning a game is $\frac{2}{3}$, and the probability of player B winning a game is $\frac{1}{3}$. The results of each game are independent of each other.  
$(1)$ Find the probability that B wins only 1 game and A wins the match.  
$(2)$ Let $X$ be the total number of games played when the match is decided. Find the probability distribution and the expected value of $X$.

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