
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 $19^{th}$ Asian Games in Hangzhou is the most prestigious international comprehensive sports event in Asia. The 19th Asian Games will be held in Hangzhou, Zhejiang from September 23 to October 8, 2023. A certain Asian Games peripheral product is very popular and in high demand, so a factory is working day and night to produce this product. The fixed cost of producing this product is $40,000, and an additional cost of $p(x)$ million is required for producing $x$ thousand units. When the production is less than 60,000 units, $p(x)=\frac{1}{2}x^{2}+x$; when the production is not less than 60,000 units, $p(x)=7x+\frac{81}{x}-\frac{63}{2}$. If the selling price of this product is $6$ yuan per unit, through market analysis, the factory can sell all the products.<br/>$(1)$ Find the function relationship of the profit $y (in million yuan)$ of selling this product with the production $x (in thousand units)$;<br/>$(2)$ At what production level will the factory maximize its profit?

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