
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
On August 29, 2023, Huawei Mate60 Pro was officially launched on Huawei Mall, becoming the world's first mass-market smartphone to support satellite calls. In recent years, the trade friction between China and the United States has been ongoing, especially the restrictions imposed by the United States on Huawei. Despite the United States' strong blockade and various pressures on other countries to resist Huawei's 5G, Huawei has not backed down. In order to further increase its market competitiveness, Huawei plans to use new technology to produce a new smartphone in 2022. Through market analysis, it is known that the production of this new phone will require a fixed cost investment of 3 million yuan for the whole year. An additional cost of f(x) million yuan is required for producing one thousand phones, where f(x) is defined as follows: 

\[ f(x) = \begin{cases} 5x + 180, & 0 < x \leq 30 \\ 602 + \frac{20000}{x(x+10)} - \frac{10000}{x}, & 30 < x \leq 110 \end{cases} \]

Market research shows that the selling price of this phone is 0.6 million yuan, and all phones produced within a year can be sold. 

$(1)$ Find the analytical expression of the profit $g(x)$ in 2022 in terms of the annual production quantity $x$ (in thousand units). 

$(2)$ For what quantity of production in thousand units in 2022 will the company make the maximum profit? What is the maximum profit?

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