
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 incurs a daily expense of $600. The relationship between the number of passengers per day $x$ (people) and the daily profit (profit = ticket revenue - expense) $y$ (dollars) is shown in the table below (the fare for each passenger remains constant):

| $x$ (people) | $\ldots $ | 200 | 250 | 300 | 350 | 400 | $\ldots $ |
|--------------|-----------|-----|-----|-----|-----|-----|-----------|
| $y$ (dollars) | $\ldots $ | -200 | -100 | 0 | 100 | 200 | $\ldots $ |

$(1)$ By observing the data in the table, to ensure no loss, how many passengers should the bus have per day?  
$(2)$ Estimate the profit when the number of passengers per day is 500.  
$(3)$ Write the expression for the relationship between $y$ and $x$.

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