
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
Teacher Wang plans to hold a math knowledge competition in the two classes he teaches, divided into individual qualifying rounds and team challenge finals. The rules for individual qualifying rounds are as follows: each person has only one chance to challenge, the computer randomly gives 5 questions, and answering 3 or more questions correctly qualifies for the next round. The rules for team challenge finals are as follows: based on the class as a unit, each class must have at least 20 participants, and the number of participants must be even. There are two ways to participate:<br/>Way 1: Divide the 2n students selected from the class into n groups of 2 students each. The computer randomly assigns the same question to two students in the same group. If at least one of the two students answers correctly, the group passes the round. If all n groups pass, the class succeeds in the challenge.<br/>Way 2: Divide the 2n students selected from the class into 2 groups of n students each. The computer randomly assigns the same question to n students in the same group. If all n students answer correctly, the group passes the round. If at least one of the two groups passes, the class succeeds in the challenge.<br/>$(1)$ Student A participates in the individual qualifying round. The probability of A answering the first three questions correctly is $\frac{1}{2}$ each, and the probability of answering the last two questions correctly is $\frac{1}{3}$ each. Find the probability of A advancing to the next round;<br/>$(2)$ In the team challenge finals, assuming that each student in a class has a constant probability of answering the given question correctly, denoted as p $(0 \lt p \lt 1)$, which way of participation should be chosen to increase the chances of the class team succeeding? Explain your reasoning.

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