
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
Carrying out garbage classification collection can reduce the amount of garbage to be processed and the processing equipment, reduce processing costs, and reduce the consumption of land resources. It has various benefits in terms of society, economy, ecology, etc., and is a major issue related to the overall ecological civilization construction. In order to popularize garbage classification knowledge, a certain school held a garbage classification knowledge exam. The exam paper only has two questions. It is known that the probability of student A answering each question correctly is $\frac{3}{4}$, the probability of student B answering each question correctly is $p$, and the results of each person answering the questions do not affect each other during the exam. It is known that the probability of exactly one of the two students A and B answering a question correctly is $\frac{5}{12}$.
$(1)$ Find the value of $p$ and the probability that both students A and B answer the questions correctly.
$(2)$ Find the probability that the sum of the number of questions answered correctly by both students is $3$.

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