
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
The 2022 Qatar World Cup is the 22nd FIFA World Cup. The competition took place from November 21 to December 18, 2022, in 12 stadiums in 7 cities in Qatar. The football association of a school organized a football knowledge exam with only two questions. It is known that the probability of Xiao Zhang answering each question correctly is $\frac{3}{4}$, the probability of Xiao Li answering each question correctly is $p$, and the results of each person's answers do not affect each other. If the probability of both of them answering the same number of questions correctly is $\frac{61}{144}$, find:<br/>
$(1)$ the value of $p$;<br/>
$(2)$ the probability of both of them answering both questions correctly.

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