
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
In a certain test, the probabilities of A, B, and C meeting the standard are respectively $\dfrac{2}{5}$, $\dfrac{3}{4}$, and $\dfrac{1}{2}$. During the test, whether A, B, and C meet the standard is independent of each other.

$(1)$ Calculate the probability that exactly $2$ people meet the standard;

$(2)$ After the test, what is the most likely number of people to meet the standard?

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