
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]


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Students A, B, and C simultaneously registered for the independent recruitment of a key university in 2010. The procedure for independent recruitment before the college entrance examination includes material review and cultural test. Only those who pass the review can participate in the cultural test, and those who pass the cultural test will be qualified for independent recruitment. Since A, B, and C each have their strengths, the probabilities of A, B, and C passing the review are 0.5, 0.6, and 0.4, respectively. After passing the review, the probabilities of A, B, and C passing the cultural test are 0.6, 0.5, and 0.75, respectively.  
(1) Calculate the probability of A, B, and C each obtaining the qualification for independent recruitment;  
(2) Let the number of people among A, B, and C who obtain the qualification for independent recruitment be $\xi$, calculate the expected value of the random variable $\xi$.

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