
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
To meet market demand, a clothing supermarket purchased a short-sleeved $T$-shirt at the beginning of June. The cost price of each shirt is $80$ yuan, and the supermarket stipulates that the selling price of each shirt must not be less than $90$ yuan. According to a survey, when the selling price is set at $90$ yuan, 600 shirts can be sold each week. It was found that for every $1$ yuan increase in the selling price of a shirt, 10 fewer shirts are sold each week.  
$(1)$ Find the function expression between the weekly sales volume $y$ (in units) and the selling price per unit $x$ yuan. (No need to specify the range of the independent variable)  
$(2)$ The clothing supermarket wants to make a profit of $8250$ yuan from the sales of this $T$-shirt each week, while also providing customers with the best value. How should the selling price of this $T$-shirt be set?  
$(3)$ The supermarket management department requires that the selling price of this $T$-shirt must not exceed $110$ yuan. Then, at what price per unit should the $T$-shirt be set so that the weekly sales profit is maximized? What is the maximum profit?

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