
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 middle school is holding a science and culture festival, and students who sign up for the history of mathematics knowledge competition need to pass two rounds of selection. The first round is a written test. If the written test is not qualified, the student cannot enter the next round of selection; if the written test is qualified, the student will proceed to the second round, which is an on-site interview. Finally, those who pass the interview will represent the grade to participate in the school-wide finals. The two rounds of selection are independent of each other. Three students, A, B, and C, have signed up for the knowledge competition. Assuming that the probabilities of A, B, and C passing the written test are $\frac{2}{3}$, $\frac{1}{2}$, and $\frac{3}{4}$ respectively, and the probabilities of passing the interview are $\frac{1}{2}$, $\frac{2}{3}$, and $\frac{1}{3}$ respectively.  
$(1)$ Find the probability that only one of the students A and B qualifies for the finals.  
$(2)$ Find the probability that at least one of the three students qualifies for the finals.

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