
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 certain school selects several students to form a math Olympiad training team. The selection process is divided into two stages. Only those who pass the first selection can proceed to the second selection. The two selection processes are independent of each other. Based on the current levels of students A, B, and C, the probabilities of A, B, and C passing the first selection are $0.5$, $0.6$, and $0.4$ respectively. For the second selection, the probabilities of A, B, and C passing are $0.6$, $0.5$, and $0.5$ respectively.  
$(1)$ Find the probability that only A passes the first selection among A and B.  
$(2)$ Find the probability that after both selections, exactly one of A, B, and C passes.

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