
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 monthly salaries (in yuan) of all employees in a certain company's sales department are as follows: $8000$; $8000$; $7000$; $6800$; $7500$; $8000$; $9500:10000$; $6800:9000$. Find:<br/>$(1)$ the average, median, and mode of the monthly salaries of all employees in the department.<br/>$(2)$ Among the three statistical measures calculated in part $(1)$, which one better reflects the level of monthly salaries in the department? Please explain your reasoning.<br/>$(3)$ Due to the company's good performance, all employees in the department received a raise of $500$ yuan in their monthly salaries. Among the "average, median, mode, and variance" of the employees' monthly salaries, which statistical measure did not change in value?

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