
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 media company plans to purchase two types of performance costumes, $A$ and $B$. It is known that each set of type $A$ costumes costs $30$ yuan more than each set of type $B$ costumes, and the number of sets of type $A$ costumes that can be purchased with 1200 yuan is the same as the number of sets of type $B$ costumes that can be purchased with 960 yuan.  
$(1)$ Find the cost of each set of type $A$ and type $B$ costumes.  
$(2)$ The company plans to purchase a total of 20 sets of type $A$ and type $B$ costumes, with the total cost not less than 2800 yuan. How many sets of type $A$ costumes should be purchased at least?

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