
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
Live streaming sales have become a popular sales method. A manufacturer conducts live sales of chestnuts on a sales platform. It is known that the cost price of chestnuts is $6$ yuan/kg, and the daily sales volume $y$ (kg) is related to the unit price $x$ (yuan/kg) by a linear function. The table below records the relevant data. Through sales, it is found that the unit price is not lower than the cost price and not higher than $32$ yuan/kg. Let $w$ (yuan) be the daily profit from selling chestnuts.

| $x$ (yuan/kg) | $10$ | $11$ | $12$ |
|---------------|------|------|------|
| $y$ (kg)      | $4000$ | $3900$ | $3800$ |

$(1)$ Find the functional relationship between the daily sales volume $y$ and the unit price $x$ and write down the range of values for the independent variable.  
$(2)$ When the unit price is set to what value, the daily profit $w$ from selling this type of chestnuts is maximized? What is the maximum profit in yuan?

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