
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 two units of a large product every day. The manufacturing cost for each unit is 2000 yuan. The probabilities of a product being a first-class product and a second-class product are 0.5 and 0.4, respectively. The selling price for each first-class product is 10000 yuan, and for each second-class product, it is 8000 yuan. If the product quality does not reach first-class or second-class standards, in addition to not recovering the costs, each product will incur a loss of 1000 yuan.
(1) Calculate the probability that, in three consecutive days of production, there is exactly one day when both produced units are first-class.
(2) Calculate the probability that if one of the two products produced in a day is first-class, the other one is also first-class.
(3) Determine the probability distribution and expectation of the daily profit $\xi$ (in yuan) from producing this product.

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