
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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A certain middle school conducted a survey to understand the satisfaction of parents with after-school extension services. They randomly selected 50 parents of students from the seventh and eighth grades for a questionnaire survey and obtained the rating data (denoted as $x$) for each parent regarding the after-school extension services. The data was then organized, described, and analyzed. The following information is provided:

$a$. The frequency distribution table of the eighth-grade parents' ratings for after-school extension services is as follows (the data is divided into 5 groups: $0\leqslant x \lt 60, 60\leqslant x \lt 70, 70\leqslant x \lt 80, 80\leqslant x \lt 90, 90\leqslant x\leqslant 100)$:

| Group            | Frequency |
|------------------|-----------|
| $0\leqslant x \lt 60$ | $2$       |
| $60\leqslant x \lt 70$ | $5$       |
| $70\leqslant x \lt 80$ | $15$      |
| $80\leqslant x \lt 90$ | $a$       |
| $90\leqslant x\leqslant 100$ | $8$       |
| Total            | $50$      |

$b$. The data of the eighth-grade parents' ratings for after-school extension services in the group $80\leqslant x \lt 90$ is arranged in ascending order, and the first 5 data points are as follows: $81$, $81$, $82$, $83$, $83$.

$c$. The average, median, and mode of the ratings of the seventh and eighth-grade parents for after-school extension services are shown in the table below:

| Grade | Average | Median | Mode |
|-------|---------|--------|------|
| Seventh | $78$ | $79$ | $85$ |
| Eighth | $81$ | $b$ | $83$ |

Based on the above information, answer the following questions:

$(1)$ $a=$______, $b=______$.

$(2)$ Give reasons why you think the after-school extension services of the grade are relatively good. (Provide reasons from at least two different perspectives.)

$(3)$ Given that a total of 600 parents of eighth-grade students participated in this rating survey, estimate approximately how many parents' ratings are not less than $80$ points.

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