
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 20th meeting of the 13th National People's Congress Standing Committee passed the "Law on the Protection of Minors," which further strengthens the responsibilities of guardians, schools, dormitory operators, and online service providers for issues such as lack of guardianship, campus bullying, tobacco and alcohol harm, and internet addiction, increasing the protection of minors. To promote the Law on the Protection of Minors, a certain middle school held a knowledge competition on the law. The rules of the competition are as follows: two people form a team, in each round of the competition, each pair answers two questions. If the number of correct answers is not less than 3, they are called an "excellent team." It is known that students A and B form a team, and the probabilities of student A and student B answering each question correctly are $p_{1}$ and $p_{2}$, respectively.
- $(1)$ If $p_{1}=\frac{3}{4}$ and $p_{2}=\frac{2}{3}$, then in the first round of the competition, find the probability that they will be an "excellent team."
- $(2)$ When $p_{1}+p_{2}=\frac{6}{5}$, and each round of the competition does not affect the others, if students A and B want to be an "excellent team" 9 times in this competition, how many rounds of competition should they theoretically participate in at least?

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