
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 wholesaler sells a product with a cost of $20$ yuan per kilogram. According to the regulations of the pricing department, the selling price of the product per kilogram should not exceed $90$ yuan. The sales volume $y$ (in kilograms) and the selling price $x$ (in yuan per kilogram) of the product satisfy a linear function relationship. The corresponding relationship is shown in the table below:

| Selling Price $x$ (yuan per kilogram) | $\ldots$ | $50$ | $60$ | $70$ | $80$ | $\ldots$ |
|---------------------------------------|----------|------|------|------|------|----------|
| Sales Volume $y$ (kilograms)          | $\ldots$ | $100$| $90$ | $80$ | $70$ | $\ldots$ |

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ At what price per kilogram does the wholesaler maximize the profit $w$ (in yuan)? What is the maximum profit at that price?

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