
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
In a certain place, four people, $A$, $B$, $C$, and $D$, successively contracted the novel coronavirus, with only $A$ having visited an epidemic area.
1. If the probabilities of $B$, $C$, and $D$ being infected by $A$ are $\frac{1}{2}$ each, what is the probability that exactly one of $B$, $C$, and $D$ is infected with the novel coronavirus?
2. If $B$ is definitely infected by $A$, for $C$, since it is difficult to determine whether they were infected by $A$ or $B$, assume that the probabilities of $C$ being infected by $A$ and $B$ are both $\frac{1}{2}$. Similarly, assume that the probabilities of $D$ being infected by $A$, $B$, and $C$ are all $\frac{1}{3}$. Under these assumptions, the number of people directly infected by $A$ among $B$, $C$, and $D$, denoted as random variable $X$, is to be determined. Find the probability distribution and the mean (expected value) of the random variable $X$.

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