
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 brand of zongzi before the arrival of the Dragon Boat Festival on May 5th. 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. For every $1 increase in the selling price 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 $x$ (in dollars).

$(2)$ At what price per box should the selling price be set to maximize the daily profit $P$ (in dollars)? What is the maximum profit?

$(3)$ To stabilize prices, the relevant management department has set a maximum selling price of $58 per box for this type of zongzi. If the supermarket wants to make a profit of at least $6000 per day, how many boxes of zongzi must be sold per day at least?

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