
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 stationery store purchased two types of commemorative books, A and B, for a total of $1200$ yuan. It is known that the cost price of type A book is $12$ yuan per book, and the cost price of type B book is $10 yuan per book. When selling, the selling price of type A book is $15$ yuan per book, and the selling price of type B book is $12$ yuan per book. After selling all the books, the store made a total profit of $270$ yuan.<br/>$(1)$ Find out how many books of type A and type B the store purchased?<br/>$(2)$ If the store purchases type A and type B books again at the original cost price, and the quantity of type A books purchased remains the same, while the quantity of type B books purchased is twice the quantity of the first purchase, and type B books are sold at the original selling price, while type A books are sold at a reduced price. When all the books are sold, to ensure that the profit from the second purchase is not less than $340$ yuan, find the minimum selling price per book for type A books.

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