
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 real estate property is holding a lottery for homebuyers, with the following rules: For homeowners who purchase the property, they can randomly draw 2 balls from box $A$, which contains 2 red balls and 2 white balls, and 2 balls from box $B$, which contains 3 red balls and 2 white balls. If all 4 balls drawn are red, the homeowner wins the first prize and receives $10,000 in renovation funds; if exactly 3 balls are red, the homeowner wins the second prize and receives $5,000 in renovation funds; if exactly 2 balls are red, the homeowner wins the third prize and receives $3,000 in renovation funds; any other outcome is considered a consolation prize, awarding $1,500 in renovation funds.  
$(Ⅰ)$ Three homeowners participate in the lottery. Find the probability that exactly one homeowner wins the second prize.  
$(Ⅱ)$ Let $X$ denote the amount of renovation funds won by a homeowner in the lottery. Find the probability distribution and the expected value of $X$.

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