
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 supermarket sells a product with a cost of $60 per kilogram. Through trial sales, it is found that the daily sales quantity $y$ (in kilograms) of this product is related to the selling price $x$ (in dollars per kilogram) by a linear function. The corresponding values of daily sales price and quantity for four days are shown in the table below:

| Selling Price $x$ (dollars per kilogram) | $65$ | $70$ | $75$ | $80$ |
|-----------------------------------------|------|------|------|------|
| Sales Quantity $y$ (kilograms)           | $70$ | $60$ | $50$ | $40$ |

$(1)$ Find the function expression between $y$ (in kilograms) and $x$ (in dollars per kilogram);  
$(2)$ To ensure a daily sales profit of $600, what should be the selling price on that day?  
$(3)$ When the selling price is set at a certain value, at what price will the daily sales profit be maximized? What is the maximum profit?

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