
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]


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In 2022, a certain company integrated funds to invest in the research and development of high-tech products, and released the first batch of 17 major technological innovation demand lists to the world, attracting more than 60 universities and institutes such as Tsinghua University and Peking University to participate. The company's product development department decided to apply a certain high-tech technology to the production of a high-tech product, planning to invest a fixed cost of 62,000 million yuan in this technology throughout the year. For each x thousand pieces of the product produced, an additional cost of F(x) million yuan is required, where F(x) is defined as follows: 

\[ F(x) = \begin{cases} 10x^2 + 100x, & 0 < x < 60 \\ 901x + \frac{8100}{x} - 21980, & x \geq 60 \end{cases} \]

Assuming that the selling price of the product to external customers is 0.9 million yuan per piece, and all the products produced within the year can be sold out. 

$(1)$ Find the function relationship $G(x)$ in million yuan for the annual profit with respect to the annual output $x$ thousand pieces. 

$(2)$ Determine the annual output in thousand pieces for which the company's profit is maximized, and calculate the maximum profit.

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