
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
The cost price of a certain product is $40$ yuan per unit. When the selling price is $50$ yuan per unit, 210 units can be sold each month. If the selling price of each product increases by $1$ yuan, then 10 fewer units are sold each month (the selling price of each unit cannot exceed $65$ yuan). Let $x$ represent the increase in selling price of each product, and $y$ represent the monthly sales volume of the product. Find:<br/>$(1)$ The functional relationship between $y$ and $x$ is: ______, and the range of the independent variable $x$ is: ______;<br/>$(2)$ What should be the selling price of each product (where $x$ is a positive integer) to maximize the monthly profit? What is the maximum monthly profit in yuan?<br/>$(3)$ If there are additional costs of $a$ yuan for each product during the sales process, and the merchant finds that the monthly sales profit decreases as $x$ increases when the selling price of each unit is not less than $58$ yuan, determine the range of values for $a$: ______.

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