
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 $50$ yuan per kilogram. Through trial sales, it was found that the daily sales quantity $y$ (in kilograms) of this product is related to the selling price $x$ (in yuan) by a linear function. The corresponding values of selling price and sales quantity for each day are shown in the table below:

| Selling Price $x$ (yuan) | $55$ | $60$ | $65$ | $70$ |
|--------------------------|------|------|------|------|
| Sales Quantity $y$ (kg)  | $70$ | $60$ | $50$ | $40$ |

$(1)$ Find the function expression between $y$ (in kilograms) and $x$ (in yuan);  
$(2)$ At what selling price should the daily sales profit be maximized? What is the maximum profit?

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