
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
A community organized a contest called "I Participate in Environmental Protection" with prizes. In a certain competition, three families, A, B, and C, answered a question about environmental protection at the same time. It is known that the probability of family A answering this question correctly is $\frac{3}{4}$, the probability that families A and C both answer incorrectly is $\frac{1}{12}$, and the probability that families B and C both answer correctly is $\frac{1}{4}$. If the correctness of each family's answer does not affect the others, find:

$(1)$ The probability that families B and C each answer this question correctly.

$(2)$ The probability that at least $2$ out of the three families A, B, and C answer this question correctly.

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