
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

...

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system
Yunmeng fish noodles is one of the traditional Han ethnic delicacies in Hubei Province, mainly produced in Yunmeng County, Hubei Province, and named after it. In 1915, Yunmeng fish noodles won the Silver Award for high-quality products in the Panama World Expo, and the products sell well in the national and international markets. This year, a fish noodle factory in Yunmeng County sells Yunmeng fish noodles on the "Rural Taobao Shop". The cost of each bag is 16 yuan. In March of this year, the shop sold 200 bags at a price of 30 yuan per bag. In order to expand sales, in April, they plan to reduce the price appropriately. According to calculations, for every 1 yuan reduction in the price per bag of noodles, the sales volume can increase by 20 bags.
  
$(1)$ When the price per bag of noodles is reduced by 5 yuan, how much profit will be made in April?
  
$(2)$ By how much should the price per bag of noodles be reduced to maximize customer benefits while allowing the manufacturer to make a profit of 2860 yuan?

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