
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 book dealer is preparing to hold a sales exhibition to increase the sales of a series of books. According to a market survey, when the selling price of each set of books is set at $x$ yuan, the sales volume can reach $\left(15-0.1x\right)$ thousand sets. Now, the publishing house, in order to cooperate with the activities of the book dealer, decides to reform the pricing by dividing the supply price of each set of books into fixed and floating prices. The fixed price is $30$ yuan, and the floating price (unit: yuan) is inversely proportional to the sales volume (unit: thousand sets) with a ratio of $10$. Assuming no other costs are considered, that is, the profit from selling each set of books $=$ selling price $-$ supply price. Find:<br/>$(1)$ The total profit obtained by the book dealer when the selling price of each set of books is set at $100$ yuan.<br/>$(2)$ At what price should each set of books be sold for the profit per set of books to be maximized.

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