
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
With the rapid development of technology, more and more businesses are transitioning to online platforms. "Livestreaming sales" has become an important promotional tool for businesses. A certain business sells a daily commodity in a livestream room with a cost price of $10$ yuan per item. Through investigation, it is found that the daily sales quantity $y$ (in items) of the commodity is related to the selling price $x$ (in yuan) by the equation $y=-10x+400$. Let the profit from selling this commodity daily be denoted as $W$ (in yuan).
  
$(1)$ If the business aims to make a profit of $1250$ yuan per day while also offering discounts to customers, what should be the selling price set at?
  
$(2)$ If the selling price is not less than $28$ yuan and at least $50$ items are sold daily, find the maximum value of $W$.

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