
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 4th China International Import Expo was held in Shanghai from November 5th to 10th, 2021. This expo showcased over 4000 new products, technologies, and services. A multinational company brought a high-end air conditioning model to the expo. Based on the expo research, a Chinese company, Enterprise A, plans to jointly produce this air conditioner with the multinational company in 2022. The production of this air conditioner is expected to require fixed costs of 2.6 million RMB for the whole year. To produce x thousand units of air conditioners, an additional investment of R million RMB is needed, where R is defined as follows:
\[ R = \begin{cases} 10x^2 + ax, & 0 \leq x < 40 \\ \frac{901x^2 - 9450x + 10000}{x}, & x \geq 40 \end{cases} \]
According to calculations, an additional investment of 40 million RMB is required when producing 10 thousand units of air conditioners. Currently, the selling price per unit of air conditioner is 0.9 million RMB. All air conditioners produced within the year can be sold in the same year.

$(1)$ Find the function relationship of the annual profit W (in million RMB) of the enterprise in 2022 with the annual output x (in thousand units).

$(2)$ For what production volume in 2022 will the enterprise achieve the maximum annual profit? What is the maximum annual profit? Note: Profit = Revenue - Cost.

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