
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
Given that a certain mobile phone company has an annual fixed cost of $\text{40}$ million USD for producing a particular phone model, an additional $\text{16}$ million USD is required for every $\text{1}$ million phones produced. Let $x$ represent the total number of phones produced annually (in millions), which are all sold. The revenue $R(x)$ (in millions of USD) per million phones is defined as follows:
$$
R(x)=
\begin{cases}
400-6x, & 0 < x\leqslant 40 \\
\frac {8000}{x}- \frac {57600}{x^{2}}, & x > 40
\end{cases}
$$
(I) Write the analytical expression for the annual profit $f(x)$ (in millions of USD) as a function of the annual production $x$ (in millions of phones);

(II) What production quantity $x$ (in millions of phones) maximizes the company's profit for this phone model? Calculate the maximum profit.

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