
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
Technological innovation is the driving force of development. A research company has launched an innovative product to the market. The cost price of the product is $40$ yuan per piece. The sales price $y$ (in yuan per piece) is related to the sales volume $x$ (in pieces) by a linear function. Some of the data is shown in the table below:

| $x$ (pieces) | $10$ | $15$ | $20$ | $\ldots$ |
|--------------|------|------|------|---------|
| $(\text{yuan per piece})$ | $58$ | $57$ | $56$ | $\ldots$ |

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ Find the analytical expression of the sales profit $w$ (in yuan) as a function of the sales volume $x$ (in pieces). At what sales volume is the sales profit maximized? What is the maximum value?  
$(3)$ In order to ensure that the sales profit is not less than $420$ yuan, find the range of values for the selling price of the product.

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