
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 household appliances department of a shopping mall decided to set an appropriate monthly sales target and reward employees who achieve the target. The household appliances department conducted statistics and analysis on the sales of 20 employees in the current month.  
Data collected (unit: ten thousand yuan):  

| $5.9$ | $9.9$ | $6.0$ | $5.2$ | $8.2$ | $6.2$ | $7.6$ | $9.4$ | $8.2$ | $7.8$ |
|-------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| $5.1$ | $7.5$ | $6.1$ | $6.3$ | $6.7$ | $7.9$ | $8.2$ | $8.5$ | $9.2$ | $9.8$ |

Data summary:  

| 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 the monthly sales amount not less than $7$ ten thousand yuan is set as the sales target, 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 amount this month is $7.5$ ten thousand yuan, higher than the average of $7.44$ ten thousand yuan, so my sales amount exceeds half of the employees, why didn't I receive a reward?" If you were the manager, please provide a reasonable explanation.

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