
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 ceramic factory is preparing to fire three different crafts, named A, B, and C. The production process must go through two firings in sequence, where the second firing can only proceed if the first one is successful. The two firing processes are independent of each other. According to the current technical level of the factory, the probabilities of A, B, and C being successfully fired in the first firing are 0.5, 0.6, and 0.4, respectively. After the second firing, the probabilities of A, B, and C being successfully fired are 0.6, 0.5, and 0.75, respectively.  
(1) Calculate the probability that exactly one craft is successfully fired after the first firing.  
(2) After both firings, let $\xi$ be the number of qualified crafts. Calculate the expected value of the random variable $\xi$.

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