
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]


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In a physical education class, students are taking a shooting test. The rule is that each student has to shoot $3$ times, and they must make at least $2$ shots to pass the test. If a student fails the test, they have to undergo $30$ additional shooting practices. It is known that student A has a probability of $\frac{1}{3}$ of making a shot each time, and the shots are independent of each other.<br/>$(1)$ Find the probability that student A passes the test;<br/>$(2)$ If student B has a probability of $\frac{1}{2}$ of making a shot each time, and the shots are independent of each other. After the test, the total number of shots that students A and B need to practice is denoted as $X$. Find the probability distribution of $X$ and the mathematical expectation $E(X)$.

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