
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 supermarket purchased a product on the eve of the New Year at a cost of $40$ yuan per box. The supermarket stipulates that the selling price per box must not be less than $45$ yuan. Based on past sales experience, it was found that when the selling price is set at $45$ yuan per box, 700 boxes can be sold per day. If the selling price is increased by $1$ yuan per box, then 20 fewer boxes are sold per day.  
$(1)$ Find the functional relationship between the daily sales volume $y$ (boxes) and the selling price per box $x$ (yuan).  
$(2)$ To make a daily profit of $6000$ yuan and provide customers with the best value, what should the selling price be?  
$(3)$ At what selling price per box will the daily profit $P$ (yuan) be maximized? What is the maximum profit?

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