
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. The cost of each box is $40. The supermarket stipulates that the selling price of each box must not be less than $45. Based on past sales experience, it was found that when the selling price is set at $45 per box, 700 boxes can be sold per day. If the selling price is increased by $1 per box, 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 $x$ (dollars).  
$(2)$ At what price per box is the daily sales profit $P$ (dollars) maximized? What is the maximum profit?

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