
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
Two students, A and B, participated in a knowledge challenge training. In the final round, there were only two questions. It is known that the probability of student A answering each question correctly is $p$, and the probability of student B answering each question correctly is $q$ ($p > q$). It is also known that the probability of at least one of them answering a question correctly is $\frac{5}{6}$, and the probability of both of them answering a question correctly is $\frac{1}{3}$.
- $(1)$ Find the values of $p$ and $q$.
- $(2)$ Find the probability that in the final round, student A answers fewer questions correctly than student B.

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