
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 3 white balls and 2 red balls of the same size and quality. In box B, there are 4 white balls and 1 red ball of the same size and quality.  
$(1)$ Take two balls from boxes A and B respectively. Let the random variable $X$ represent the number of red balls among the balls taken. Find the probability distribution and the expected value of $X$.  
$(2)$ First, take two balls from box A and put them into box B. Then, take two balls from box B. Find the probability of drawing two white balls from box B.

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