
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 fruit store has a linear relationship between the daily sales volume $y$ (in kilograms) of a fruit and the selling price $x$ (in dollars per kilogram). Some of the data is shown in the table below:

| Selling Price $x$ (dollars per kilogram) | $6$ | $8$ | $10$ |
|-----------------------------------------|-----|-----|------|
| Daily Sales Volume $y$ (kilograms)      | $20$| $18$| $16$ |

$(1)$ Find the functional relationship between the daily sales volume $y$ and the selling price $x$ of this fruit.<br/>$(2)$ If the price per kilogram of this fruit is limited to the range of $6$ dollars to $12$ dollars, find the range of daily sales volume of this fruit.<br/>$(3)$ Given that the purchase price of this fruit is $4$ dollars per kilogram, find the maximum gross profit that can be obtained from selling this fruit without exceeding a daily sales volume of $10$ kilograms. (Assumption: Gross profit = Revenue - Cost Price)

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