
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 certain brand clothing company has designed a new garment, with a cost price of $60 ($ per piece). Before a large-scale launch, a market survey was conducted to understand the popularity of the style and the relationship between the daily sales volume $y$ (pieces) and the selling price $x ($ per piece). Some information is shown in the table below:

| Selling Price $x ($ per piece) | $80$ | $90$ | $100$ | $110$ |
|--------------------------------|------|------|-------|-------|
| Daily Sales Volume $y$ (pieces) | $240$| $220$| $200$ | $180$|

$(1)$ If $y$ and $x$ satisfy a linear function relationship, please write down the analytical expression of the function directly (do not need to write the range of the independent variable $x$);  
$(2)$ If the company wants to make a profit of $8000$ yuan per day and wants to give as much benefit to customers as possible, how should they price the garment?  
$(3)$ In order to help children in poor mountainous areas, the company has decided to donate $10$ yuan to Hope Primary School for every garment sold. How should the company price the garment to maximize the daily profit? (Profit denoted by $w$)

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