
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 and B classes of a certain school jointly held a "Classic Reading" competition. $10$ students were randomly selected from class A and class B respectively, and the competition scores of these students were collected, organized, and analyzed. The following is some of the information provided.<br/>【Data Collection】<br/>Scores of $10$ students in class A: $85$, $78$, $86$, $79$, $72$, $91$, $79$, $71$, $70$, $89$<br/>Scores of $10$ students in class B: $85$, $80$, $77$, $85$, $80$, $73$, $90$, $74$, $75$, $81$<br/>【Data Organization】<br/><table class="edittable"><tbody><tr><td width="65">Class</td><td width="65">$70\leqslant x \lt 80$</td><td width="65">$80\leqslant x \lt 90$</td><td width="65">$90\leqslant x \lt 100$</td></tr><tr><td>Class A</td><td>$6$</td><td>$3$</td><td>$1$</td></tr><tr><td>Class B</td><td>$4$</td><td>$5$</td><td>$1$</td></tr></tbody></table>【Data Analysis】<br/><table class="edittable"><tbody><tr><td width="65">Class</td><td width="65">Mean</td><td width="65">Median</td><td width="65">Mode</td><td width="65">Variance</td></tr><tr><td>Class A</td><td>$80$</td><td>$a$</td><td>$b$</td><td>$51.4$</td></tr><tr><td>Class B</td><td>$80$</td><td>$80$</td><td>$80$,$85$</td><td>$c$</td></tr></tbody></table>【Problem Solving】Based on the above information, answer the following questions:<br/>$(1)$ Fill in the blanks: $a=$______,$b=$______,$c=$______；<br/>$(2)$ Based on the information in 【Data Analysis】, determine which class has better scores, and briefly explain the reason;<br/>$(3)$ Class A has a total of $45$ students, and Class B has a total of $40$ students. According to the competition rules, students with scores of $80$ and above can win prizes. Estimate the total number of students from these two classes who can win prizes?

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