
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 successful hosting of the Winter Olympics has greatly inspired people's enthusiasm for building a strong sports nation and has sparked a wave of physical exercise among young people. In order to understand the situation of "physical fitness standards" among all students in a school, 40 students were randomly selected from the 1000 students in the senior year to participate in the "physical fitness standards" test. It was stipulated that those with a predicted score of less than 60 points would be considered "unqualified", otherwise qualified. If the number of "unqualified" students in the senior year does not exceed 5% of the total number of students, then the senior year's physical fitness standards are considered "qualified"; otherwise, they are considered "unqualified", and the senior year students need to undergo additional training. These 40 students were randomly divided into two groups, Group A with 24 students and Group B with 16 students. After the prediction, the statistical analysis of the predicted scores for each group is as follows: the average score of Group A is 70 with a standard deviation of 4; the average score of Group B is 80 with a standard deviation of 6. (The final results are all rounded to the nearest integer).

1. Find the average score $\overline{x}$ and standard deviation $s$ of the test scores of these 40 students.
2. Assuming that the predicted scores of senior students' physical fitness standards follow a normal distribution $N(\mu, \sigma^2)$, use the sample mean $\overline{x}$ as an estimate $\widehat{\mu}$ of $\mu$ and the sample standard deviation $s$ as an estimate $\widehat{\sigma}$ of $\sigma$. Use the estimates to determine whether the predicted physical fitness standards of senior students are "qualified".
3. To enhance the fun, in the skipping test item of the physical fitness standards, students can challenge the strong players in the sports specialty class. Each challenge match adopts a best-of-seven format. The scoring rules are as follows: Winning with a score of 4:0 or 4:1 earns 4 points for the winning team member and 0 points for the losing team member; Winning with a score of 4:2 or 4:3 earns 3 points for the winning team member and 1 point for the losing team member. Assuming that the athlete Wang Qiang has a probability of winning each game of $\frac{2}{3}$, find the probability that Wang Qiang wins the first 3 games in this round of matches and earns 3 points. 

Given:
- The variance of $n$ numbers is ${s^2}=\frac{1}{n}\sum_{i=1}^n{{{({x_i}-\overline{x})}^2}}$.
- If a random variable $Z\sim N(\mu, \sigma^2)$, then $P\left(\mu -\sigma\ \ \lt Z \lt \mu +\sigma \right)=0.6826$, $P\left(\mu -2\sigma\ \ \lt Z \lt \mu +2\sigma \right)=0.9544$, $P\left(\mu -3\sigma\ \ \lt Z \lt \mu +3\sigma \right)=0.9974$.

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