
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

...

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It is known that a certain mobile phone brand in the United States has an annual fixed cost of $400,000 for producing a certain model of mobile phone, and an additional investment of $160,000 is required for every 10,000 units produced. Suppose the company produces and sells $x$ ten thousand units of this model within a year, and the revenue per ten thousand units, $R(x)$, in ten thousand dollars, is given by
\[R(x)=\begin{cases}
400-6x, & 0 < x\leqslant 40 \\
\dfrac{7400}{x}- \dfrac{40000}{{x}^{2}}, & x > 40
\end{cases}\]

(1) Write the analytical expression for the annual profit $W$ (in ten thousand dollars) as a function of the annual production $x$ (in ten thousand units).

(2) At what annual production level (in ten thousand units) does the company achieve the maximum profit from producing this model of mobile phone? And calculate the maximum profit.

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