
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
It is known that the annual fixed cost of producing a certain model of phone by an electronics company is $400,000. An additional $160,000 is required for every 10,000 units produced. Let the company produce a total of $x$ thousand units of this phone in a year and sell them all. The sales revenue per thousand units is denoted by $R(x)$ in million US dollars, where $R(x)=\left\{{\begin{array}{l}{400-6x,0≤x≤40}\\{\frac{{7400}}{x}-\frac{{40000}}{{{x^2}}},x＞40}\end{array}}\right.$.<br/>$(1)$ Write the function expression of the annual profit $W$ (in million US dollars) in terms of the annual production quantity $x$ (in thousand units) (Profit = Sales Revenue - Cost);<br/>$(2)$ At what production quantity in thousand units does the company maximize the profit in the production of this phone? Calculate the maximum profit.

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