
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 high school selects 20 students to form a math Olympiad training team, with 8 students from Grade 11, 7 students from Grade 12, and 5 students from Grade 13.  
$(1)$ If 3 students are randomly selected 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 Grade 11.  
$(2)$ The school wants to assess the members of the math Olympiad training team. The assessment rules are as follows: there are 4 questions in total, the first 2 questions are worth 1 point each for a correct answer and 0 points for a wrong answer, and the last 2 questions are worth 2 points each for a correct answer and 0 points for a wrong answer. Students who accumulate a score of at least 5 points are considered excellent students. It is known that for a student named Zhang, the probability of answering each of the first 2 questions correctly is $\frac{2}{3}$, and the probability of answering each of the last 2 questions correctly is $\frac{1}{2}$. It is also known that the answers to each question do not affect each other. Let $X$ denote Zhang's cumulative score in this assessment. Find the probability distribution and mathematical expectation of $X$, and determine the probability that Zhang will be awarded the title of an excellent student in this assessment.

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