
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 factory, in order to set a reasonable price for a newly developed product, conducted a trial sale of the product at the planned prices. By analyzing the sales data of 5 days, the following information was obtained:

| Unit Price (yuan per item) | $30$ | $34$ | $38$ | $40$ | $42$ |
|----------------------------|------|------|------|------|------|
| Sales Volume (items)       | $40$ | $32$ | $24$ | $20$ | $16$ |

(1) Calculate the average sales amount over these 5 days (sales amount = unit price $\times$ sales volume);

(2) Through analyzing the data in the table above, it was found that there exists a linear relationship between the sales volume $y$ (items) and the unit price $x$ (yuan per item). Find the functional relationship of $y$ with respect to $x$ (no need to specify the range of the independent variable);

(3) It is expected that in future sales, the sales volume and unit price will still follow the relationship in (2), and the cost of this product is $20$ yuan per item. In order for the factory to maximize profit, what should be the unit price of the product?

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