
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. The cost price of each shirt is $50$ yuan, and it is stipulated that the selling price of each shirt is not less than the cost price. According to a market survey, the monthly sales volume $y$ (in units) and the selling price per unit $x$ (in yuan) satisfy a linear function relationship $y=-20x+2600$.
$(1)$ 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 the price of this shirt be set?
$(2)$ The price department stipulates that the profit per unit of this shirt should not exceed $30\%$ of the cost price. If the total monthly profit of this shirt is $w$ yuan, what selling 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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