
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 high-quality fruit at a cost price of $40$ yuan per kilogram. If the selling price is $50$ yuan per kilogram, then $500$ kilograms can be sold in a month. If the selling price increases by $1$ yuan based on $50$ yuan per kilogram, the monthly sales volume decreases by $10$ kilograms.  
$(1)$ When the monthly profit is $8750$ yuan, what is the selling price per kilogram of fruit?  
$(2)$ When the selling price per kilogram of fruit is what value, the monthly profit is maximized?

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