
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
A certain county's famous traditional local products "bean sprouts" and "dried tofu" are renowned nationwide for their "rich bean fragrance, green healthiness," and are beloved by consumers. It is known that the purchase price for 2 units of bean sprouts and 3 units of dried tofu is $240, and the purchase price for 3 units of bean sprouts and 4 units of dried tofu is $340.  
$(1)$ Find the purchase price for each unit of bean sprouts and dried tofu.  
$(2)$ A specialty store plans to purchase bean sprouts and dried tofu totaling 200 units for no more than $10,440, with the quantity of bean sprouts not less than $\frac{3}{2}$ of the quantity of dried tofu. What are the possible purchasing schemes for this specialty store?  
$(3)$ If the selling price for each unit of bean sprouts is $80 and each unit of dried tofu is $55, under the conditions of $(2)$, how should the store purchase the products to maximize profit, and what is the maximum profit?

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