
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 fruit wholesaler sells apples at a cost of $40$ yuan per box. A market survey found that if the price per box is set at $50$ yuan, an average of $90$ boxes are sold per day. It was also observed that for every $1$ yuan increase in price, the average daily sales decrease by $3$ boxes.<br/>$(1)$ Find the functional relationship between the average daily sales $y$ (in boxes) and the selling price $x$ (in yuan per box);<br/>$(2)$ Find the functional relationship between the average daily sales profit $w$ (in yuan) and the selling price $x$ (in yuan per box);<br/>$(3)$ If the price department stipulates that the selling price per box must not exceed $55$ yuan, at what price per box can the wholesaler achieve maximum profit?

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