
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 store sells a type of student backpack. It is known that the cost price of this backpack is $60$ yuan each. Market research shows that if the backpack is priced at $110$ yuan, the sales volume per month is 300 units. If the price is reduced by $1$, the sales volume increases by 10 units per month. Let the unit price of this backpack be $x$ yuan, and the monthly sales profit be $w$ yuan.  
$(1)$ Find the functional relationship between $w$ and $x$.  
$(2)$ At what unit price should this backpack be sold to maximize the monthly sales profit? What is the maximum profit?  
$(3)$ The store is dedicated to public welfare and decides to donate $1750$ yuan to the Hope Project from the monthly profit of this backpack. In order to ensure that the remaining profit after donation is not less than $12000$ yuan per month, find the range of the unit price $x$.

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