
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 Education Bureau of a certain city uses the knowledge content of the "Study in China" learning platform as the basis, requiring the directly-administered schools in the city to use the "Yushiban" mobile client to conduct the "Reviewing the Top Twenty" national knowledge competition. The Education Bureau randomly selected $10$ teachers from two schools for testing (full marks $10$ points) and organized the data as follows:  
Collecting data:  
Test scores of $10$ teachers selected from School One: $9.1$, $7.8$, $8.5$, $7.5$, $7.2$, $8.4$, $7.9$, $7.2$, $6.9$, $9.5$  
Test scores of $10$ teachers selected from School Two: $9.2$, $8.0$, $7.6$, $8.4$, $8.0$, $7.2$, $8.5$, $7.4$, $7.5$, $8.2$  
Analyzing data: The relevant statistical data of the two groups are as follows (it is stipulated that $9.0$ points and above are excellent):  

|         | Mean  | Median | Variance | Excellent Rate |
|---------|-------|--------|----------|----------------|
| School One | $8.0$ | $7.85$ | $0.666$  | $c$            |
| School Two | $8.0$ | $b$    | $0.33$   | $10\%$         |

Problem solving: Based on the above information, answer the following questions:  
$(1)$ If a frequency distribution table is drawn for the score segments, then the frequency $a$ of the score segment $0\leqslant x \lt 8.0$ in School One is ______;  
$(2)$ Fill in the blanks: $b=$______, $c=$______;  
$(3)$ If School One has a total of $280$ teachers and School Two has a total of $350$ teachers, estimate the total number of teachers from these two schools who achieved excellent scores in the competition;  
$(4)$ Based on the data above, please analyze and evaluate the competition scores of the teachers from School One and School Two. (Write two points)

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