
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
In order to motivate employees, the manager of the home appliance department of a shopping mall decided to set an appropriate monthly sales target and reward employees who achieve the target. The home appliance department conducted statistics and analysis on the sales of 20 employees in the month.  
Data collection (unit: ten thousand yuan):  
$5.0, 9.9, 6.0, 5.2, 8.2, 6.2, 7.6, 9.4, 8.2, 7.8$  
$5.1, 7.5, 6.1, 6.3, 6.7, 7.9, 8.2, 8.5, 9.2, 9.8$  
Data organization:  

| Sales Amount / Ten Thousand Yuan | $5\leqslant x \lt 6$ | $6\leqslant x \lt 7$ | $7\leqslant x \lt 8$ | $8\leqslant x \lt 9$ | $9\leqslant x \lt 10$ |
|----------------------------------|----------------------|----------------------|----------------------|----------------------|----------------------|
| Frequency                        | $3$                  | $5$                  | $a$                  | $4$                  | $4$                  |

Data analysis:  

| Mean   | Mode  | Median |
|--------|-------|--------|
| $7.44$ | $8.2$ | $b$    |

Problem solving:  
$(1)$ Fill in the blanks: $a=4$, $b=7.7$.  
$(2)$ If setting the sales target for the month to be no less than $7$ ten thousand yuan, then ____ employees will receive rewards.  
$(3)$ After the manager analyzed the data, rewards were given to half of the employees. Employee A found the manager and said, "My sales this month is $7.5$ ten thousand yuan, higher than the average of $7.44$ ten thousand yuan, so my sales exceed half of the employees, why didn't I get a reward?" If you were the manager, please provide a reasonable explanation.

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