
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 product by a company is $400,000, and an additional $16 is required for each product produced. Let the company produce a total of $x$ thousand products in a year and sell them all. The sales revenue per thousand products is denoted by $R(x)$ million, and it is known that $R(x)=\left\{{\begin{array}{l}{400-6x,0＜x≤40}\\{\frac{{7400}}{x}-\frac{{40000}}{{{x^2}}},x＞40}\end{array}}\right.$.<br/>$(1)$ Find the analytical expression of the profit $W (in million dollars)$ as a function of the annual output $x$ (in thousand pieces);<br/>$(2)$ For how many thousand pieces of output? The company obtains the maximum profit in the production of this product, and find the maximum profit.

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