
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 supermarket purchased a brand of yuanxiao (a type of traditional Chinese dessert) on the eve of the Lantern Festival. The cost of each box is $20$ yuan, and the supermarket stipulates that the selling price of each box must not be less than $25$ yuan. Based on past sales experience, it was found that when the selling price is set at $25$ yuan per box, the supermarket can sell $250$ boxes per day. For every increase of $1$ yuan in the selling price per box, the supermarket sells $10$ boxes less per day. <br/>$(1)$ Find the functional relationship between the daily sales volume $y$ (boxes) and the selling price per box $x$ (yuan). <br/>$(2)$ At what price per box should the selling price be set to maximize the daily sales profit $P$ (yuan)? What is the maximum profit? <br/>$(3)$ To stabilize prices, the relevant management department stipulates that the selling price per box of this type of yuanxiao must not exceed $38$ yuan. If the supermarket wants to make a profit of at least $2000$ yuan per day, how many boxes of yuanxiao must be sold per day at least?

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