
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 stationery store purchased a batch of learning supplies at a unit price of $12. According to the regulations of the relevant department, the selling price should not be lower than the purchase price and should not exceed $1.5$ times the purchase price. Through analyzing the sales situation, it was found that the daily sales quantity $y$ (in pieces) and the selling price $x$ (in dollars) satisfy a linear function relationship. When $x=15$, $y=50$; when $x=17$, $y=30$.  
$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ When should the selling price of these learning supplies be set to maximize daily profit, and what is the maximum profit in dollars?

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