
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 organizes five students, named A, B, C, D, and E, to participate in a testing activity at a university. There are two different testing schemes, labeled A and B, and each student randomly chooses one scheme for testing. The probability of passing the test for scheme A is $\frac{2}{3}$, and for scheme B is $\frac{1}{2}$. The results of each student's test are independent of each other.<br/>$(1)$ If students A, B, and C choose scheme A, and students D and E choose scheme B, find the probability that exactly three students pass the test;<br/>$(2)$ If the average number of students passing the test is not less than 3, determine the number of students choosing scheme A for testing.

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