
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
On October 1, 2019, which marked the 70th anniversary of the founding of the People's Republic of China, a merchant purchased a batch of commemorative shirts for 3200 yuan. After they were put on the market, they were in high demand. The merchant then purchased a second batch of the same type of commemorative shirts for 7200 yuan. The quantity purchased in the second batch was twice that of the first batch, but each shirt was 10 yuan more expensive.  
$(1)$ What was the unit price of the first batch of commemorative shirts purchased by the merchant?  
$(2)$ If the two batches of commemorative shirts are sold at the same price, and the remaining 20 shirts are sold at an 20% discount off the price, ensuring that the total profit from selling all the shirts is at least 3520 yuan (without considering other factors), what is the minimum selling price per shirt?

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