
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 factory produces 2 units of a large product each day, with each product having a cost input of 2000 yuan. The probability that a product is of first-class quality is 0.5, and the probability that it is of second-class quality is 0.4. Each first-class product sells for 10,000 yuan, and each second-class product sells for 8,000 yuan. If the product quality does not meet first or second-class standards, there is a loss of 1,000 yuan per product produced, in addition to the unrecoverable cost.

(1) Find the probability that, over 3 consecutive days of production, exactly one day has both products being first-class.

(2) Given that on a certain day, one of the two products is a first-class product, find the probability that the other product produced on that day is also a first-class product.

(3) Determine the probability distribution and expectation of the daily profit $\xi($ yuan $)$.

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