
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
Not only is Nanshan District in Shenzhen a beautiful coastal city, but it is also a city full of youth and vitality, a city of technology and innovation, ranking first in the national "Top 100 Districts" for five consecutive years. The drone manufacturer "DJI Innovation Technology" in this district is well-known worldwide. The company's drone accessories sales department currently has two types of accessories, A and B, with their purchase prices and selling prices as shown in the table. With $15,000, you can purchase 50 units of product A and 25 units of product B. (Profit = selling price - purchase price) <br/>$(1)$ Find the value of the purchase price $a$ for product A. <br/>$(2)$ If the sales department purchases a total of 300 units of product A and product B, according to market sales analysis, the number of units of product B purchased must be at least twice the number of units of product A. How should the purchase be made to maximize the profit from this sale? What is the maximum profit in yuan? <br/><table><tbody><tr><td width="123">Type</td><td width="85">Product A</td><td width="83">Product B</td></tr><tr><td>Purchase Price (yuan/unit)</td><td>$a$</td><td>$80$</td></tr><tr><td>Selling Price (yuan/unit)</td><td>$300$</td><td>$100$</td></tr></tbody></table>

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