
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
On the eve of the Mid-Autumn Festival, a cake shop purchased a brand of mooncakes at a cost of $60$ yuan per box. The cake shop stipulates that the selling price per box must not be less than $70$ yuan. Based on past sales experience, it was found that when the selling price is set at $70$ yuan per box, 500 boxes can be sold per day. For every increase of $1$ yuan in the selling price per box, 20 fewer boxes are sold per day. Please answer the following questions:<br/>$(1)$ If the selling price of each box of mooncakes is increased by $20$ yuan, how many boxes can be sold per day and what is the sales profit?<br/>$(2)$ Let the daily sales profit be $y$ yuan, and the selling price per box be increased by $x$ yuan $(x$ is an integer$)$. Find the functional expression between $y$ and $x$;<br/>$(3)$ At what price per box will the total daily sales profit be maximized? What is the maximum profit?

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