
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
Selecting astronauts can be said to be "one in ten thousand". To pass, one must go through five stages: visual inspection, initial inspection, re-inspection, cultural examination, and political review. If students A, B, and C from a certain school all successfully pass the first two stages, the probabilities of passing the re-inspection for students A, B, and C are $0.5$, $0.6$, and $0.75$ respectively. The probabilities of passing the cultural examination for them are $0.6$, $0.5$, and $0.4$ respectively. Due to their good performance, they can all pass the political review. If the outcomes of the last three stages are independent of each other, find:<br/>
$(1)$ The probability that student A will be recruited as an air force pilot;<br/>
$(2)$ The probability that exactly one of students A, B, and C will pass the re-inspection.

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