
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 Mathematical Olympiad Competition is a traditional intellectual competition aimed at selecting outstanding talents through competition, promoting the intellectual development of young people. Many excellent universities have requirements for the results of middle school students in the Olympiad competition as part of their foundation program. Therefore, various middle schools attach great importance to this. A certain middle school selected a total of 15 students through an exam to form a math Olympiad training team, including 7 students from the first year, 6 students from the second year, and 2 students from the third year.  
$(1)$ If the school randomly selects 3 students from the math Olympiad training team to participate in a math Olympiad, find the probability that exactly 2 of the selected 3 students are from the first year.  
$(2)$ Now the school wants to assess the members of the math Olympiad training team through an exam consisting of 3 questions. In the test, answering at least 2 questions correctly is considered passing. It is known that the probability of Zhang answering each question correctly is $\frac{1}{2}$, the probability of Wang answering each question correctly is $\frac{2}{3}$, and each student's answers to the questions do not affect each other. Let $X$ be the number of students who pass the exam, find the probability distribution and mathematical expectation of $X$.

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