
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
Since 2020, more and more businesses have been transitioning to online platforms, and "livestreaming sales" has become an important promotional tool for businesses. A certain business sells a daily commodity in live streaming with a cost price of $10 per item. Through investigation, it is found that the daily sales quantity $y$ (in items) of this commodity and the selling price $x$ (in dollars) satisfy $y=-10x+400$. The profit of selling this commodity daily is denoted as $W$ (in dollars).
- $(1)$ Find the functional relationship between $W$ and $x.
- $(2)$ If the business wants to make a profit of $1250$ dollars daily and reduce inventory, what should be the selling price?
- $(3)$ If the selling price is not less than $28$ dollars 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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