
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
The computer exam consists of two parts: a theoretical exam and a practical operation exam. Each part of the exam is graded as "pass" or "fail". Those who pass both parts of the exam will be considered as having passed the computer exam and will be awarded a "certificate of passing". The probabilities of passing the theoretical exam for A, B, and C are $\dfrac{4}{5}$, $\dfrac{3}{4}$, and $\dfrac{2}{3}$ respectively, and the probabilities of passing the practical operation exam are $\dfrac{1}{2}$, $\dfrac{2}{3}$, and $\dfrac{5}{6}$ respectively. The outcomes of all exams are independent of each other.

$(1)$ If A, B, and C all take both the theoretical and practical operation exams at the same time, who has the highest probability of obtaining the "certificate of passing"?

$(2)$ Calculate the probability that exactly two out of A, B, and C obtain the "certificate of passing" after taking both the theoretical and practical operation exams.

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