
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 19th Asian Games will be held in Hangzhou, China from September 23, 2023, to October 8, 2023. A high school in Hangzhou organized a knowledge competition titled "Asian Games in My Heart" for the first-year students. The number of participants from Class 1, Class 2, Class 3, and Class 4 are as follows:

| Class | 1    | 2    | 3    | 4    |
|-------|------|------|------|------|
| Number of Students | 30   | 40   | 20   | 10   |

From the students who signed up, 10 students were selected to participate in the competition using stratified sampling. Each participating student randomly answered 4 out of 10 preset questions, and a student who answered at least 3 questions correctly would receive a prize. It is assumed that the answering situation of each student is independent.

$(1)$ Find the number of students participating from each class.  
$(2)$ If Xiaozhang from Class 2 is selected to participate in the competition and answers exactly 3 questions incorrectly out of the 10 preset questions, let $X$ be the number of questions he answers correctly. Find the probability distribution and the expected value of $X$.  
$(3)$ If each student from Class 1 has a probability of $\frac{1}{3}$ of answering each question correctly, find the probability that at least one student from Class 1 will receive a prize in the competition.

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