
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 2022 Beijing Winter Olympics, a certain online store purchased two types of Bing Dwen Dwen keychains, $A$ and $B$, directly from the factory. The purchase price and selling price are shown in the table below: (Note: Profit = Selling Price - Purchase Price)

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

$(1)$ During the first purchase, the online store spent $850$ yuan to buy a total of $30$ keychains of types $A$ and $B$. Find the number of keychains purchased for each type.  
$(2)$ After selling out the keychains purchased in the first round, the online store plans to purchase a total of $80$ keychains of types $A$ and $B$ in the second round (with unchanged purchase and selling prices), and the total purchase price should not exceed $2200$ yuan. How should the purchase plan be designed to maximize the sales profit, and what is the maximum sales profit?  
$(3)$ As the Winter Olympics is coming to an end, the online store plans to adjust the selling price of the $B$ keychains. If sold at the original price, an average of $4$ keychains can be sold per day. After investigation, it was found that for each $1$ yuan reduction in price, an additional $2$ keychains can be sold per day. What should be the selling price per keychain to ensure an average daily sales profit of $90$ yuan for the $B$ keychains?

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