
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
The Lantern Festival is a traditional festival in China. People have the custom of eating tangyuan. As the Lantern Festival approaches, a certain supermarket plans to purchase a batch of tangyuan for sale.
$(1)$ If the supermarket purchases a total of 1000 bags of two brands, brand $A$ and brand $B$, and the brand $A$ tangyuan is 20 bags more than twice the brand $B$ tangyuan, how many bags of brand $A$ and brand $B$ tangyuan should be purchased?
$(2)$ The purchaser of the supermarket found that one bag of brand $B$ tangyuan is $6$ yuan more expensive than one bag of brand $A$ tangyuan, and the cost of purchasing 5 bags of brand $A$ tangyuan is the same as purchasing 3 bags of brand $B$ tangyuan. What are the purchase prices of brand $A$ and brand $B$ tangyuan?

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