
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]


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To celebrate the "May Fourth" Youth Day, a class organized a patriotic knowledge competition, with teams A and B participating, each consisting of $3$ people. Each person answers a question, scoring $1$ point for a correct answer and $0$ points for a wrong answer. Assuming that the probability of each person in team A answering a question correctly is $\frac{1}{3}$, and the probabilities for team B are $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$ respectively, with no influence between the correctness of answers within each team. <br/>$(1)$ Find the probabilities of team A scoring a total of $3$ points and $1$ point respectively; <br/>$(2)$ Find the probability of team A scoring a total of $2$ points and team B scoring a total of $1$ point.

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