
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
BingDwenDwen is the mascot of the 2022 Beijing Winter Olympics. It combines the image of a panda with a super-powered ice crystal shell. The head shell is modeled after a snow sports helmet, decorated with a colorful halo, and the overall image resembles an astronaut. As the Winter Olympics approaches, BingDwenDwen dolls are selling very well. Xiao Li selected two BingDwenDwen dolls, A and B, on a certain online store, and decided to purchase and sell them from that store. The purchase price and selling price of the two dolls are as follows:

|          | A doll | B doll |
|----------|--------|--------|
| Purchase Price (¥/each) | $20$ | $15$ |
| Selling Price (¥/each) | $25$ | $18$ |

$(1)$ Xiao Li purchased 100 dolls of A and B for a total of ¥1650. Find out how many dolls of each type Xiao Li purchased.

$(2)$ When Xiao Li made a second purchase, the online store stipulated that the quantity of A dolls purchased should not exceed half of the quantity of B dolls purchased. Xiao Li plans to purchase a total of 100 dolls of both types. How should Xiao Li design the purchase plan to maximize profit, and what is the maximum profit?

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