
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 wholesaler purchases a certain type of vegetable at a cost of $24$ yuan per box and sells it to a retail supermarket. It is known that the listed price of this vegetable is $45$ yuan per box, and the actual selling price is not less than 80% of the listed price. The wholesaler analyzed the sales situation and found that the sales volume $y$ (in boxes) of this vegetable satisfies a linear function relationship with the selling price $x$ (in yuan per box). The table shows two sets of corresponding values:

| Selling Price $x$ (yuan per box) | ... | $35$ | $38$ | ... |
|----------------------------------|-----|------|------|-----|
| Sales Volume $y$ (boxes)         | ... | $130$| $124$| ... |

$(1)$ If the selling price of this vegetable is $42$ yuan per box on a certain day, then the sales volume of this vegetable on that day is ______ boxes.  
$(2)$ Can the wholesaler make a profit of $1320$ yuan on a certain day by selling this vegetable? If yes, find the selling price on that day. If not, explain the reason.  
$(3)$ The wholesaler is running a promotion where for every box of this vegetable purchased, a $6$ yuan worth of potatoes is given for free. What should be the selling price of this vegetable to maximize the daily profit, and what is the maximum daily profit in yuan?

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