
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
"Xiaolongxia" is the flagship product of our county's characteristic agriculture and is widely cultivated in Nan County. It is estimated that a certain village had a cultivation area of 100 mu in 2020, and by 2022, the cultivation area of the village reached 196 mu.  
$(1)$ Calculate the average growth rate of the cultivation area of "Xiaolongxia" in this village over these two years.  
$(2)$ A survey conducted by a breeding household found that when the selling price of "Xiaolongxia" is 20 yuan per kilogram, 200 kilograms can be sold per day. For every 1 yuan decrease in price, an additional 50 kilograms can be sold per day. In order to promote and publicize, the breeding household decided to reduce the price for promotion and reduce the inventory. It is known that the average cost of "Xiaolongxia" is 12 yuan per kilogram. If the goal is to ensure a daily profit of 1750 yuan, how much should the price be reduced by?

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