
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 brand of cars is preparing to give out free ice cream to customers during a car exhibition. They plan to wholesale ice cream from the factory. It is known that the fixed cost of the factory in this order is $20,000, the maximum production limit is 80,000 units, and in addition, the cost will increase by $0.5$ million for every 10,000 ice creams produced. The sales revenue $f(x)$ for every $x$ million ice creams satisfies the relationship $f(x)=-\frac{{x}^{3}}{2}+\frac{9}{4}ax^{2}+\frac{x}{2}$ (unit: million dollars, where $a$ is a constant). If the factory sells 20,000 ice creams, the profit is $120,000.<br/>$(1)$ Let the profit from selling $x$ million ice creams be $g(x)$ (unit: million dollars), find the analytical expression of $g(x)$;<br/>$(2)$ For what sales volume of this order is the profit of this factory maximized? And find the maximum profit.

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