
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 retailer purchased a batch of trendy clothing for $48,000 and sold it through an online platform. Due to good market conditions, they purchased the same type of clothing for $100,000 a second time. The quantity purchased the second time was twice the quantity purchased the first time, and the cost per item increased by $10.

$(1)$ How many items of this type of clothing did the retailer purchase the first time, and at what price per item?

$(2)$ After selling the first batch of clothing, the retailer found that if they sold each item for $300, they could sell an average of 80 items per day. If the selling price decreased by $10, they could sell 20 more items. Following this trend, when selling the second batch of clothing at a promotional discount, they made a profit of exactly $3600 per day. What should the selling price be?

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