
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
During the Spring Festival, a certain online store purchased two types of products, $A$ and $B$, from the factory. The purchase price and selling price are shown in the table below: (Note: Profit $=$ Selling Price $-$ Purchase Price)

| Category | $A$ Product | $B$ Product |
|----------|------------|------------|
| Purchase Price (元/each) | $30$ | $25$ |
| Selling Price (元/each) | $45$ | $37$ |

$(1)$ The online store first purchased $A$ and $B$ products for a total of $30$ items with $850$ yuan. Find the number of items purchased for each product.  
$(2)$ After selling out the products purchased the first time, the online store plans to purchase $80$ items of $A$ and $B$ products (with unchanged purchase and selling prices) for a total purchase price not exceeding $2200$ yuan. How should the purchase plan be designed to maximize sales profit, and what is the maximum sales profit?  
$(3)$ As the Spring Festival draws to a close, the online store plans to adjust the selling price of the $B$ product. If sold at the original price, an average of $4$ items can be sold per day. After investigation, it was found that for every $1$ yuan reduction in price, an additional $2$ items can be sold per day. What should be the selling price per item of the $B$ product to make the average daily sales profit for the $B$ product $90$ yuan?

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