
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
Chinese outstanding traditional culture has a long history and is the crystallization of Chinese civilization. "The Mathematical Classic of Sunzi" and "The Mathematical Classic of Zhou Bi" are relatively popular ancient Chinese arithmetic books with many interesting and straightforward problems. The unit price of "The Mathematical Classic of Sunzi" in a bookstore is $\frac{3}{4}$ of the unit price of "The Mathematical Classic of Zhou Bi". Buying "The Mathematical Classic of Sunzi" for $600$ yuan is equivalent to buying $5$ more copies than buying "The Mathematical Classic of Zhou Bi".<br/>$(1)$ Find the unit prices of the two books in yuan.<br/>$(2)$ For the "3.14 Mathematics Festival" event, a certain school plans to purchase a total of $80$ copies of these two books from the bookstore, with the quantity of "The Mathematical Classic of Zhou Bi" not less than half of the quantity of "The Mathematical Classic of Sunzi". Due to the large purchase quantity, the bookstore offers an 20% discount on both books. Find the minimum cost when purchasing a certain number of copies of each book.

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