
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]


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On March 1, 2021, the State Council Information Office held a press conference, during which the Ministry of Industry and Information Technology proposed five measures for the development of chips, further encouraging domestic technology giants to increase their investment in technology research and development. According to market research, the annual fixed cost for a digital product company to produce a certain type of sports bracelet is 500,000 yuan, and an additional 200,000 yuan is required for every 10,000 units produced. If the company produces x thousand units of this sports bracelet in a year and sells them all, the average sales revenue per 10,000 units is R(x) thousand yuan, where R(x) is defined as follows: 

\[ R(x) = \begin{cases} 100-kx, & 0<x\leq 20 \\ \frac{2100}{x}-\frac{9000k}{x^2}, & x>20 \end{cases} \]

When the company produces and sells 50,000 units of this sports bracelet in a year, the annual profit is 3 million yuan. 
- (1) Find the value of k and write the analytical expression of the profit W (in thousand yuan) in terms of the annual output x (in ten thousand units).
- (2) For what annual output in ten thousand units does the company maximize its profit in the production of this sports bracelet? What is the maximum profit?

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