
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
In response to the national call for "mass entrepreneurship and innovation," after graduating from university, student Xiao Wang decided to start a business using his professional knowledge. After conducting market research, it was found that producing a certain small electronic product requires an annual fixed cost of $20,000. For each production of $x$ thousand units, an additional variable cost of $C(x)$ thousand dollars is required. When the annual output is less than 80,000 units, $C(x)=\frac{1}{3}{x^2}+2x$ (thousand dollars); when the annual output is not less than 80,000 units, $C(x)=7x+\frac{100}{x}-37$ (thousand dollars). The selling price of each product is $6. Assuming that all the products produced by Xiao Wang are sold out in the same year.<br/>$(Ⅰ)$ Write the analytical expression of the function of annual profit $P(x)$ (thousand dollars) in terms of the annual output $x$ (thousand units);<br/>$(Ⅱ)$ For how many thousand units of annual output, Xiao Wang will obtain the maximum profit in the production of this product? What is the maximum profit?

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