
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
A certain school organized a "Welcome the 20th National Congress, Forge Ahead on a New Journey" essay competition for 100 students each from the eighth and ninth grades. The competition scores (in points) of 10 randomly selected students from each grade are collected and organized as follows:  
Collected data:  
Eighth grade: $74$, $76$, $79$, $81$, $84$, $86$, $87$, $90$, $90$, $93$.  
Ninth grade: $76$, $81$, $81$, $83$, $84$, $84$, $84$, $85$, $90$, $92$.  
Organized data:  
The competition scores for the eighth and ninth grades are organized into score ranges as follows:  

|          | $70\leqslant x \lt 80$ | $80\leqslant x \lt 90$ | $90\leqslant x \lt 100$ |
|----------|------------------------|------------------------|------------------------|
| Eighth grade | $a$ | $4$ | $3$ |
| Ninth grade | $1$ | $7$ | $2$ |

Analyzing the data:  
The average, median, mode, and variance of the scores for the eighth and ninth grades are organized as follows:  

|          | Average | Median | Mode | Variance |
|----------|---------|--------|------|----------|
| Eighth grade | $84$ | $b$ | $90$ | $36.4$ |
| Ninth grade | $84$ | $84$ | $c$ | $18.4$ |

Problem solving:  
Based on the above information, answer the following questions:  
$(1)$ Write down $a=$______, $b=$______, $c=______.  
$(2)$ According to the data analysis above, which grade, eighth or ninth, has better competition scores? Please provide one reason.  
$(3)$ If a competition score of at least $85$ points is considered "excellent," estimate the number of students in these two grades who achieved an "excellent" score.

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