
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 clothing wholesale market sells a type of shirt, with a purchase price of $50$ yuan per shirt. It is stipulated that the selling price of each shirt is not lower than the purchase price. According to a market survey, the monthly sales volume $y$ (in units) and the selling price $x$ (in yuan) per unit satisfy a linear function relationship. Some of the data is shown in the table below:

| Selling Price $x$ (yuan/unit) | $60$ | $65$ | $70$ |
|-------------------------------|------|------|------|
| Sales Volume $y$ (units)       | $1400$ | $1300$ | $1200$ |

$(1)$ Find the functional expression between $y$ and $x$; (no need to find the range of the independent variable $x$)<br/>$(2)$ The wholesale market wants to make a profit of $24000$ yuan from the sales of this type of shirt each month, while also providing customers with affordable prices. How should they price this type of shirt?<br/>$(3)$ The price department stipulates that the profit per shirt should not exceed $30\%$ of the purchase price. If the total monthly profit from this type of shirt is $w$ yuan, what price should be set to maximize the profit? What is the maximum profit?

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