
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
Table tennis is the national sport of China. Chinese players have won most of the championships in the world table tennis competition, and have even swept all the championships of the entire event multiple times. Table tennis is also deeply loved by people. Table tennis balls mainly come in two colors, white and yellow. The International Table Tennis Federation uses star ratings to represent the level of the balls. The star rating represents the quality level, with higher stars indicating better quality. The highest level is "☆☆☆", which is a three-star ball designated for professional competitions by the International Table Tennis Federation. Two-star balls are suitable for major domestic competitions and national team professional training, while one-star balls are suitable for amateur competitions or fitness training. A box contains 9 table tennis balls, including 2 three-star "☆☆☆" white balls, 4 one-star "☆" white balls, 1 three-star "☆☆☆" yellow ball, and 2 one-star "☆" yellow balls. 

$(1)$ Draw two balls one by one without replacement. Let event $A=\{$first ball is white$\}$ and event $B=\{$second ball is a three-star ball$\}$. Find $P\left(B\right)$, $P\left(AB\right)$, $P\left(B|A\right)$, and determine whether events $A$ and $B$ are independent.  
$(2)$ Draw balls one by one without replacement until a white ball is drawn, and let the number of three-star balls drawn be a random variable $X$. Find the probability distribution table of random variable $X$ and its expectation.

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