
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 fruit store sells a type of fruit. The cost price of this fruit is $40$ yuan per kilogram. According to market research, the weekly sales quantity $y$ (in kilograms) of this product is a linear function of the selling price $x$ (in yuan per kilogram). Some of the data is shown in the table below:

| Selling Price $x$ (yuan per kilogram) | $45$ | $60$ | $70$ | $75$ |
|---------------------------------------|------|------|------|------|
| Weekly Sales Quantity $y$ (kilograms) | $110$ | $80$ | $60$ | $50$ |

$(1)$ Find the function expression between $y$ and $x$;  
$(2)$ At what selling price per kilogram can the maximum profit be obtained each week? What is the maximum profit in yuan?  
$(3)$ Due to certain reasons, the store's cost price increases by $m$ yuan per kilogram $(m > 0)$. Through sales records, it is found that when the selling price is greater than $76$ yuan per kilogram, the weekly profit decreases as the selling price increases. Determine the range of values for $m$.

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