
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]


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The cost price of a certain product is $40$ yuan per unit, and the selling price is $50$ yuan per unit. The product can be sold at a rate of $210$ units per month. If the selling price of each product increases by $1$ yuan, then the number of units sold per month decreases by $10$ (the selling price of each product cannot exceed $65$ yuan). Let the selling price of each product increase by $x$ yuan $(x$ is a positive integer$)$. The monthly profit from sales is denoted as $y$ yuan. <br/>$(1)$ Find the functional relationship between $y$ and $x$, and directly state the range of values for the independent variable $x$; <br/>$(2)$ At what price should each product be sold to maximize the monthly profit? What is the maximum monthly profit?

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