
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 ninth-grade math interest group conducted a market survey and found that the monthly sales volume of a certain type of sportswear is a linear function of the selling price, with the following information:<br/>

| Selling Price (元/each) | $100$ | $110$ | $120$ | $130$ | $\ldots $ |
|--------------------------|-------|-------|-------|-------|-----------|
| Monthly Sales Volume (pieces) | $200$ | $180$ | $160$ | $140$ | $\ldots $ |

It is known that the cost price of this sportswear is $60$ yuan per piece, and the selling price is denoted as $x$ yuan.<br/>

$(1)$ Please express the profit of selling each piece of this sportswear in terms of $x$:① The profit of selling each piece of this sportswear is (______) yuan;<br/>
$(2)$ Find the linear function relationship between the monthly sales volume $y$ and the selling price $x$:<br/>
$(3)$ Assuming the monthly profit from selling this sportswear is $W$ yuan, then at what price is the monthly profit maximized? What is the maximum profit?

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