
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 games in a row will win the match. If after playing 5 games there is no consecutive wins, the player with more wins will be declared the winner. Assuming that the probability of player A winning a game is $\frac{2}{3}$ and the probability of player B winning a game is $\frac{1}{3}$, and that the outcomes of each game are independent of each other.<br/>$(1)$ Find the probability that B wins only 1 game and A wins the match;<br/>$(2)$ Let $X$ be the total number of games played when the match is decided, find the probability distribution and expectation of $X$.

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