
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
In box A, there are 2 white balls and 2 red balls, and in box B, there is 1 white ball and 3 red balls. Now, 2 balls are randomly selected from box A and placed into box B, then 1 ball is randomly selected from box B.  
$(1)$ Let the random variable $X$ represent the number of red balls drawn from box A. Find the distribution table and the expected value of $X$.  
$(2)$ Find the probability of drawing 1 red ball from box B.

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