
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 "Elite Plan" started in 2013, jointly organized and implemented by the China Association for Science and Technology and the Ministry of Education. By 2023, it has trained over 6000 outstanding high school students with innovative potential. To select and train candidates, a university selects outstanding students from high schools in Chongqing to participate in summer camps for mathematics, physics, and chemistry.

$(1)$ If there are exactly 5 students from school A among the 12 students in the mathematics group, and 3 students are selected from these 12 students, where $\xi$ represents the number of students selected from school A, find the distribution table and mathematical expectation of $\xi$.

$(2)$ At the opening ceremony of the summer camp, the physics group held a subject knowledge quiz competition. The rules are as follows: two students form a team, in each round of the quiz, each person answers two questions. If the team answers at least 3 questions correctly, they win the round. It is known that students A and B form a team, and the probabilities of A and B answering each question correctly are $p_{1}$ and $p_{2}$, respectively, with ${p_1}+{p_2}=\frac{4}{3}$. Assuming that the results of each round of questions are independent, if students A and B want to win 6 rounds in this quiz, how many rounds of competition should they theoretically participate in?

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