
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
The mascots "Bing Dwen Dwen" and "Shuey Rhon Rhon" are beloved by everyone as the mascots of the Beijing Winter Olympics and Paralympics. A flagship store sells "Bing Dwen Dwen" plush toys for a total of $24000$ yuan and "Shuey Rhon Rhon" plush toys for a total of $8000$ yuan. The selling price of "Bing Dwen Dwen" is $40$ yuan more than that of "Shuey Rhon Rhon," and the quantity of "Bing Dwen Dwen" sold is twice the quantity of "Shuey Rhon Rhon." <br/>$(1)$ Find the selling prices of "Bing Dwen Dwen" and "Shuey Rhon Rhon" respectively in yuan. <br/>$(2)$ It is known that the costs of "Bing Dwen Dwen" and "Shuey Rhon Rhon" are $100$ yuan per unit and $60$ yuan per unit, respectively. In January 2022, these two plush toys continue to sell well, so the flagship store purchased a total of 800 of these toys. The quantity of "Shuey Rhon Rhon" does not exceed three times the quantity of "Bing Dwen Dwen," and the total purchase price of these two plush toys does not exceed $57600$ yuan. To thank new and old customers, the flagship store decided to reduce the price of "Bing Dwen Dwen" by $10\%$ before selling. If all the plush toys purchased in January are sold out, how many "Bing Dwen Dwen" should the flagship store purchase in order to maximize the monthly sales profit, and what is the maximum profit?

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