
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]


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To welcome the arrival of the campus science and technology festival, the school's science and technology society intends to purchase two types of models, A and B, for assembly. It is known that the total price of 3 sets of model A is equal to the total price of 2 sets of model B. If purchasing 1 set of model A and 2 sets of model B costs a total of $80. <br/>$(1)$ Find the unit price of model A and model B, respectively. <br/>$(2)$ Now planning to use a budget of $19320, without exceeding the budget, to purchase a total of 800 sets of these two models. The quantity of model B should not be less than $\frac{2}{3}$ of the quantity of model A. How many different purchasing schemes are there for the two models? How many sets of model B should be purchased to minimize the total cost?

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