
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 region, the number of children $X$ in a family is as follows:

| $X$ | $1$ | $2$ | $3$ | $0$ |
|-----|-----|-----|-----|-----|
| $P$ | $\frac{6}{15}$ | $\frac{6}{15}$ | $\frac{2}{15}$ | $\frac{1}{15}$ |

The probability that each child is a boy or a girl is both $\frac{1}{2}$, and they are independent of each other. Then the probability that a family has more boys than girls is ______.

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