
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 supermarket purchases a stationery item at a price of $10$ yuan each. When selling the item, the selling price is not less than the purchase price and not more than $19$ yuan. After a market survey, it was found that there is a linear relationship between the daily sales quantity $y$ (units) of the item and the selling price $x$ (yuan). Some of the data is shown in the table below:

| Selling Price $x/$yuan | $\ldots $ | $12$ | $13$ | $14$ | $\ldots $ |
|-----------------------|---------|------|------|------|---------|
| Daily Sales Quantity $y/$units | $\ldots $ | $36$ | $34$ | $32$ | $\ldots $ |

$(1)$ Write down the function relationship between $y$ and $x$ directly.  
$(2)$ If the supermarket makes a profit of $192$ yuan from selling this item every day, what is the selling price?  
$(3)$ Let the profit from selling this item daily be $w$ (yuan). At what selling price is the daily profit maximized? What is the maximum profit?

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