
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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The 4th China International Import Expo was held in Shanghai from November 5th to 10th, 2021. This expo showcased over 400 new products, technologies, and services. A multinational company brought a high-end air conditioning model to the expo. Based on the expo survey, a Chinese company, Company 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 yuan for the whole year. To produce x thousand units of air conditioners, an additional investment of R million yuan is needed, where R is defined as follows:
\[ R=\begin{cases} 10x^2+ax, & 0\leq x<40 \\ \frac{901x^2-9450x+10000}{x}, & x\geq 40 \end{cases} \]
It was calculated that when producing 10 thousand units of air conditioners, an additional investment of 40 million yuan is required. Currently, the selling price per unit of air conditioner is 0.9 million yuan, and all air conditioners produced within the year can be sold.

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

$(2)$ For what production volume in 2022 will the company achieve the maximum annual profit? What is the maximum annual profit?

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