
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
During the World Cup, the school organized a World Cup football knowledge competition, which includes both single-choice questions and multiple-choice questions (both with four options):  
$(1)$ In the knowledge competition, if a person, let's call them A, does not know the answer to a single-choice question, they will guess randomly. It is known that the probabilities of A knowing the answer to a single-choice question and A randomly guessing the answer to a single-choice question are $\frac{2}{3}$ and $\frac{1}{3}$, respectively. The question is: when A is answering a certain single-choice question and gets it right, what is the probability that A knows the answer to this single-choice question?  
$(2)$ When A is answering a multiple-choice question and has no idea about the correctness of the four options, they can only randomly choose based on their experience. The probabilities of A choosing one option, two options, and three options are $0.5$, $0.3$, and $0.2$, respectively. It is known that each multiple-choice question has two or three correct options out of the four options. If all options are correct, A gets 5 points; if some options are correct, A gets 2 points; if there are incorrect choices, A gets 0 points. For a certain multiple-choice question with three correct options, let X be the score A gets for answering this multiple-choice question. Find the probability distribution and the expected value of X.

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