
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
The 19th Asian Games will be held in Hangzhou in September 2023. The theme of this Asian Games is "green, intelligent, thrifty, and civilized", showcasing the beauty of Hangzhou's ecology and culture, fully leveraging the traction effect of international major events on urban development, thereby promoting rapid economic development. During the preparation, a company brought in an intelligent device for procurement negotiations and decided to put a large number of them into the local market. It is known that the annual fixed R&D cost of this device is 500,000 yuan, and an additional 800,000 yuan is required for producing ten thousand units. The company will produce x ten thousand units of this device within a year and sell them all. When 0 < x ≤ 20, the relationship between the annual sales revenue (in ten thousand yuan) per ten thousand units and the annual production x (ten thousand units) satisfies the equation: t = 180 - 2x; when x > 20, the relationship between the annual sales revenue (in ten thousand yuan) per ten thousand units and the annual production x (ten thousand units) satisfies the equation: t = 70 + 2000/x - 9000/(x(x+1)).  
$(1)$ Write the analytical expression of the annual profit y (in ten thousand yuan) in terms of the annual production x (ten thousand units) (Profit = Sales Revenue - Cost);  
$(2)$ For how many ten thousand units of annual production will the company obtain the maximum annual profit? And calculate the maximum profit.

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