
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 Logistics Service Center of the North Lake Stone Room monitors the quality of the school cafeteria on the third floor by conducting two cafeteria service quality sampling surveys every semester. The specific method of each survey is to randomly survey 50 dining teachers and students, asking them to rate the cafeteria service quality. Based on their feelings, the teachers and students select a score from 0 to 100 to rate the cafeteria service quality. Based on the ratings of these 50 teachers and students on the cafeteria service quality, a frequency distribution table is drawn. The following table is the frequency distribution table drawn based on the results of the second sampling survey of this semester, with the sample data grouped as $\left[40,50\right)$, $\left[50,60\right)$, $\ldots \ldots $, $\left[90,100\right]$.

| Score | $\left[40,50\right)$ | $\left[50,60\right)$ | $\left[60,70\right)$ | $\left[70,80\right)$ | $\left[80,90\right)$ | $\left[90,100\right]$ |
|-------|----------------------|----------------------|----------------------|----------------------|----------------------|------------------------|
| Frequency | 0.04 | 0.06 | 0.22 | 0.28 | 0.22 | 0.18 |

$(1)$ Replace each group of data with the midpoint value and estimate the average score of the first three groups in the frequency distribution table.  
$(2)$ The school randomly selects 3 students and the principal Tian to have lunch together every week. Each time, Principal Tian will assess the cafeteria service quality through these 3 students. Principal Tian's approach is to let the students choose between "poor rating," "average rating," and "good rating." If there is a "poor rating" or "no good rating," Principal Tian will immediately instruct the deputy principal in charge of logistics to personally inspect the cafeteria service. Using the frequency distribution table of the 50 sampled teachers and students as the basis for estimating the population, estimate the probability using frequency estimation. Assume that among the students who have lunch with Principal Tian this week, those who rate between $\left[40,60\right)$ will give a "poor rating," those who rate between $\left[60,80\right)$ will give an "average rating," and those who rate between $\left[80,100\right]$ will give a "good rating." It is known that students will independently give evaluations based on their feelings without being influenced by other factors. Estimate the probability that Principal Tian will instruct the deputy principal in charge of logistics to personally inspect the cafeteria service quality this week.

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