
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 wholesaler sells a product with a cost price of $10$ dollars per unit. Market research shows that the weekly sales quantity $y$ (in units) of the product is related to the selling price $x$ (in dollars per unit) by a linear function $y=-10x+400$. The weekly profit $w$ of selling this product is in dollars.  
$(1)$ Find the functional relationship between $w$ and $x$.  
$(2)$ At what selling price per unit will the weekly profit be maximized?  
$(3)$ In order to revitalize funds and reduce inventory, the business wants to ensure that the weekly sales quantity of this product is not less than $180$ units. If the weekly profit from selling this product is $2000$ dollars, what is the weekly sales quantity of this product?

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