
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
To actively create a "demonstration school for the prevention and control of myopia in children and adolescents in the city" and cultivate students' good eye habits, a certain school conducted a competition on correct eye knowledge this semester. $20$ student answer sheets were randomly selected and their scores (denoted by $x$, unit: points) were recorded as follows:  
$86\ \ 82\ \ 90\ \ 99\ \ 98\ \ 96\ \ 90\ \ 100\ \ 89\ \ 83$  
$87\ \ 88\ \ 81\ \ 90\ \ 93\ \ 100\ \ 96\ \ 100\ \ 92\ \ 100$  
Organize the data:  
<table>
<tr>
<td width="139" align="center">$80\leqslant x \lt 85$</td>
<td width="139" align="center">$85\leqslant x \lt 90$</td>
<td width="139" align="center">$90\leqslant x \lt 95$</td>
<td width="139" align="center">$95\leqslant x\leqslant 100$</td>
</tr>
<tr>
<td align="center">$3$</td>
<td align="center">$4$</td>
<td align="center">$a$</td>
<td align="center">$8$</td>
</tr>
</table>  
Analyze the data:  
<table>
<tr>
<td width="138" align="center">Mean</td>
<td width="138" align="center">Median</td>
<td width="138" align="center">Mode</td>
</tr>
<tr>
<td align="center">$92$</td>
<td align="center">$b$</td>
<td align="center">$c$</td>
</tr>
</table>  
Based on the above information, answer the following questions:  
$(1)$ Write down the values of $a$, $b$, and $c$ directly from the table;  
$(2)$ $2700$ students from the school participated in the knowledge competition. Estimate the number of people with scores not less than $90$ points;  
$(3)$ Choose one quantity from the median and mode, and explain its significance in this question.

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