
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
Chinese New Year is a traditional festival in China. Every year after New Year's Day is the peak shopping period. In January 2023, a fruit merchant purchased two types of Red Fuji apples, $A$ and $B$, from farmers. The purchase price of type $A$ Red Fuji apples is $28$ yuan per piece, and the selling price is $42$ yuan per piece. The purchase price of type $B$ Red Fuji apples is $22$ yuan per piece, and the selling price is $34$ yuan per piece. (Note: Profit $=$ Selling Price $-$ Purchase Price)  
$(1)$ For the first time, the fruit shop purchased a total of $30$ pieces of type $A$ and type $B$ Red Fuji apples for $720$ yuan. Find the number of pieces purchased for each type of Red Fuji apples.  
$(2)$ After selling out the Red Fuji apples purchased for the first time, the fruit shop plans to purchase a total of $80$ pieces of type $A$ and type $B$ Red Fuji apples for the second time (with unchanged purchase and selling prices), and the total purchase cost should not exceed $2000$ yuan. How should the purchase plan be designed to maximize the sales profit, and what is the maximum sales profit?  
$(3)$ Towards the end of the Chinese New Year, the fruit shop found a large quantity of type $B$ Red Fuji apples remaining and decided to adjust the selling price of type $B$ Red Fuji apples. If sold at the original price, an average of $4$ pieces can be sold per day. After investigation, it was found that for every $1$ yuan reduction in price, an average of $2$ more pieces can be sold per day. In order to reduce the inventory quickly, at what price per piece should the selling price be set so that the average daily sales profit for type $B$ Red Fuji apples is $90$ yuan?

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