
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
A certain shopping mall is holding a promotional event where customers can participate in a lucky draw after purchasing a certain amount of goods. Each draw is made from Box A containing 4 red balls and 6 white balls, and Box B containing 5 red balls and 5 white balls. If both balls drawn are red, the customer wins first prize; if only one ball is red, the customer wins second prize; if no red balls are drawn, the customer does not win any prize.  
$(1)$ Find the probability of a customer winning a prize after one draw.  
$(2)$ If a customer has 3 chances to participate in the draw, and let X be the number of first prizes the customer wins in the 3 draws, find the distribution table and the expected value of X.

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