
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 certain bus has a monthly expenditure of $6000$ yuan, with a fare of $2$ yuan per person. Let $x$ represent the number of people who ride the bus each month. The difference between the monthly income and expenditure is $y$ yuan.  
$(1)$ Write the relationship between $y$ and $x$ ______;  
$(2)$ Fill in the table based on the relationship between $y$ and $x$  
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
x (\text{people}) & \ldots & 1000 & 1800 & 2500 & 3100 & 3800 & 4500 & \ldots \\
\hline
y (\text{yuan}) & \ldots & \_\_\_\_ & \_\_\_\_ & \_\_\_\_ & \_\_\_\_ & \_\_\_\_ & \_\_\_\_ & \ldots \\
\hline
\end{array}
\]
$(3)$ How many people need to ride the bus each month for it not to operate at a loss?

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