
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 item. After investigation, the salesperson found that when the selling price is $100$ yuan, the monthly sales volume is 50 items. Furthermore, for every $2$ yuan decrease in the selling price, an additional 10 items can be sold per month, with the condition that the selling price cannot be lower than the cost.<br/>$(1)$ Find the functional relationship between the monthly sales volume $y$ (items) and the selling price $x$ (yuan) of the product. (No need to determine the range of the independent variable.)<br/>$(2)$ If the monthly sales profit of the product is $4000$ yuan and to provide customers with more benefits, what should be the selling price?<br/>$(3)$ The salesperson at the supermarket discovered that when the monthly sales volume of the product exceeds a certain quantity, the profit obtained may decrease. To maximize the monthly profit, what should be the selling price of the product?

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