
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
There is a single-choice question with four options and a multiple-choice question with four options, where each multiple-choice question has two or three correct options. According to the scoring rules: getting all correct answers earns 5 points, getting some correct answers earns 2 points, and getting any wrong answer earns 0 points. Therefore, when answering the multiple-choice question, Xiao Ming may choose one, two, or three options, but not all four options.<br/>$(1)$ If Xiao Ming does not know the correct answer to the single-choice question, he will make a random guess. It is known that Xiao Ming knows the correct answer to the single-choice question and the probability of guessing randomly are both $\frac{1}{2}$. After he completes the single-choice question, given that he answered it correctly, find the probability that he knows the correct answer to the single-choice question.<br/>$(2)$ Assuming that when Xiao Ming answers the multiple-choice question, based on his previous experience, the probability of choosing one option is $\frac{1}{2}$, choosing two options is $\frac{1}{3}$, and choosing three options is $\frac{1}{6}$. It is known that the multiple-choice question has only two correct options, Xiao Ming has no idea about the correctness of the four options, so he can only randomly choose based on his experience. Let $X$ represent the score Xiao Ming gets after completing the multiple-choice question. Find:<br/>① $P\left(X=0\right)$;<br/>② The distribution and expected value of $X$.

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