
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 middle school, in order to create the "most beautiful campus library", has newly purchased a batch of books. The price of each popular science book is $1.2$ times the price of each literature book. It is known that the number of literature books purchased by the school for $1200$ yuan is $10$ more than the number of popular science books purchased with the same amount of money.<br/>$(1)$ What is the price of each literature book and each popular science book purchased by the school?<br/>$(2)$ In the new semester, the school plans to purchase a total of $1000$ books of popular science and literature, with the number of popular science books purchased not less than $\frac{2}{3}$ of the number of literature books. How should the purchase be made to minimize the cost? What is the minimum cost in yuan?

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