
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 automotive parts factory in Suizhou City is a specialized manufacturer of a certain component. The fixed cost of production each year is 4 million yuan, and an additional investment of 1.6 million yuan is required for every 10,000 units produced. The factory has a good reputation, and the product quality is superior, leading to a high demand in the market. If the factory produces $x$ ten-thousand units of the component each year, with a revenue of $R(x)$ ten-thousand yuan for each ten-thousand units sold, and $R(x)$ is given by the function:
$$
R(x) = \begin{cases} 
400 - 6x, & \text{if } 0 < x \leq 40 \\
\frac{7400}{x} - \frac{40000}{x^2}, & \text{if } x > 40
\end{cases}
$$

1. Write down the expression for the annual profit function in terms of the annual production $x$ (in ten-thousand units).
2. At what annual production level (in ten-thousand units) does the factory obtain the maximum profit? And calculate the maximum profit.

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