
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 order to promote knowledge of garbage classification, a school held a garbage classification knowledge exam. The exam paper only had two questions. It is known that student A has a probability of $p$ of answering each question correctly, while student B has a probability of $q$ of answering each question correctly, with $p > q$. It is also known that the probability of both student A and student B answering a question correctly is $\frac{1}{3}$, and the probability of exactly one of them answering a question correctly is $\frac{1}{2}$.
$(1)$ Find the values of $p$ and $q$;
$(2)$ Find the probability that both students answer $3$ questions correctly.

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