
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 number of children in the families $A$, $B$, $C$, $D$, and $E$ are shown in the table below:

|         | $A$ | $B$ | $C$ | $D$ | $E$ |
|---------|-----|-----|-----|-----|-----|
| Boys    | $0$ | $1$ | $0$ | $1$ | $1$ |
| Girls   | $0$ | $0$ | $1$ | $1$ | $2$ |

$(1)$ If a girl is randomly selected from these children, given that the selected child is a girl, find the probability that the girl is from family $E$;  
$(2)$ If three families are selected randomly from these $5$ families, and $X$ represents the number of families where girls outnumber boys, find the probability distribution and expectation of $X$.

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