
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 sells a type of stationery. The cost price of each item is $2. According to long-term sales data, it is found that when the selling price of this stationery is $3 per item, 500 items can be sold per day. If the selling price increases by $0.1, the sales volume will decrease by 10 items. The Price Bureau stipulates that the selling price cannot exceed 240% of the cost price.

$(1)$ If this type of stationery is to achieve a daily profit of $800, what should be the selling price per item?

$(2)$ How should it be priced to maximize the daily profit of this type of stationery? What is the maximum profit?

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