
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
Qiqi selected two dolls, $A$ and $B$, in a certain online store and decided to purchase and sell them from the store. The purchase price and selling price of the two dolls are as shown in the table below:

|         | $A$ doll | $B$ doll |
|---------|----------|----------|
| Price   | $8$      | $8$      |
| Purchase Price | $20$ | $15$ |
| Selling Price | $28$ | $20$ |

$(1)$ Qiqi spent $550$ yuan to purchase a total of $30$ dolls of $A$ and $B$ for the first time. The number of dolls purchased for $A$ and $B$ are ______ each.

$(2)$ When Qiqi purchased the dolls for the second time, the online store stipulated that the quantity of $A$ dolls purchased should not exceed half of the quantity of $B$ dolls purchased. Qiqi plans to purchase a total of $30$ dolls of the two types. How should she design the purchase plan to maximize profit? What is the maximum profit?

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