
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 19th Asian Games will be held in Hangzhou, China from September 23 to October 8, 2023. The mascot of this Asian Games is a set of three robots: "Cong Cong" representing the World Heritage Site Liangzhu Ancient City Ruins, "Lian Lian" representing the World Heritage Site West Lake, and "Chen Chen" representing the World Heritage Site Grand Canal. A public welfare team plans to hold an exhibition and sale of the Hangzhou Asian Games mascots, and all the profits will be used for the construction of community sports facilities. It is known that the cost price of each set of mascots is $\left(50+a\right)$ yuan, where $a$ is inversely proportional to the quantity purchased. When purchasing 10,000 sets, $a$ is 9 yuan. According to market research, when the selling price of each mascot is set at $x$ yuan $\left(x \lt 100\right)$, the sales volume can reach $(10-\frac{x}{10})$ thousand sets. If the other expenses for the exhibition are 10,000 yuan, and all the purchased items are sold out.<br/>$(1)$ When the selling price of each mascot is set at 70 yuan, how much profit can be obtained in total in ten thousand yuan?<br/>$(2)$ For what value of $x$, the net profit per mascot is maximized?

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