
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 group of college students seized a business opportunity and purchased a batch of dried fruits to be packaged into small, nutritionally balanced packages for sale. The cost of each bag is $3. During the sales period, it was found that the daily sales quantity $y$ (bags) was related to the selling price $x$ (in dollars) by a linear function. Some of the data is shown in the table below, where $3.5 \leq x \leq 5.5$. Additionally, there are other expenses of $80$ dollars per day.

| Selling Price $x$ (in dollars) | $3.5$ | $5.5$ |
|--------------------------------|-------|-------|
| Sales Quantity $y$ (bags)      | $280$ | $120$ |

1. Write down the function relationship between $y$ and $x$ directly.
2. If a profit of $160$ dollars is made per day, what is the selling price?
3. Let the daily profit be $w$ dollars. For what selling price will the daily profit be maximized, and what is the maximum profit?

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