
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
The grape variety "Early Black Treasure" is a high-quality new variety developed by the Shanxi Academy of Agricultural Sciences and is widely planted in Shanxi Province. By the end of the year $2018$, a certain grape planting base in a city had already planted "Early Black Treasure" on $100$ mu of land, and by the end of the year $2020$, the planting area of "Early Black Treasure" had reached $196$ mu.  
$(1)$ Calculate the average annual growth rate of the planting area of "Early Black Treasure" in this base over the two years.  
$(2)$ Market research has found that when the selling price of "Early Black Treasure" is $20$ yuan per kilogram, $200$ kilograms can be sold per day. For every $1$ yuan decrease in the selling price, an additional $50$ kilograms can be sold per day. In order to reduce inventory, the base decided to reduce the price for promotion. It is known that the average cost of "Early Black Treasure" in this base is $12$ yuan per kilogram. If the base wants to make a profit of $1750$ yuan per day from selling "Early Black Treasure", how much should the selling price be reduced by?

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