
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 school offers a school-based course on programming intelligent robots and has purchased two models of robots, $A$ and $B$. The unit price of model $A$ is $200$ yuan more than the unit price of model $B$. The school spent $2000$ yuan to purchase the same quantity of model $A$ as they did with $1200$ yuan to purchase model $B.<br/>$(1)$ Find the unit prices of model $A$ and model $B.<br/>$(2)$ The school is preparing to purchase a total of $40$ units of model $A$ and model $B$, with the quantity of model $B$ not exceeding three times the quantity of model $A$. The vendor offers a discount of $20\%$ for both models. What is the minimum cost and how many units of model $A$ and model $B should be purchased to achieve this minimum cost?

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