
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. After analyzing the sales data for $5$ days, the following data was obtained:<br/>

| Price (元/Item) | $30$ | $34$ | $38$ | $40$ | $42$ |
|-----------------|------|------|------|------|------|
| Quantity (Items) | $40$ | $32$ | $24$ | $20$ | $16$ |

$(1)$ By analyzing the data in the table above, it was found that there is a linear relationship between the quantity $y$(items) and the price $x$(元/item). Find the functional relationship of $y$ with respect to $x$ (no need to write the range of the independent variable);<br/>
$(2)$ It is expected that in future sales, the quantity and price will still have the relationship in $(2)$, and the cost of the product is $20$元/item. In order for the factory to maximize profit, what should be the price of the product?<br/>
$(3)$ To ensure that the actual sales volume of the product in the trial sale is not less than $30$ items, and the profit obtained by the factory is not less than $400元, please directly write down the range of values for the price $x$.

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