
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 company has developed a new product with an annual fixed cost of $150 million for production. To produce $x$ thousand units, an additional cost of $t$ million is required, where $t$ is defined as:
$$
t=
\begin{cases}
\frac{1}{3}x^2 + 10x, & 0 < x < 80 \\
51x + \frac{10000}{x} - 1550, & x \geq 80
\end{cases}
$$
Each unit of the product is sold at a price of $500, and the entire production can be sold on the market.

(I) Write the analytical expression of the annual profit $y$ (in millions) as a function of the annual production $D$ (in thousands of units).

(II) What is the annual production volume (in thousands of units) that maximizes the company's profit from producing this new product?

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