
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 total expenditure for producing a certain quantity of goods is called the production cost, which can be represented as a function of the quantity of goods. It is known that when a company produces a certain product in the quantity of $x$ pieces, the cost function is  
$C(x) = 20 + 2x + 0.5x^2$ (in ten thousand yuan). If the income from selling one piece of this product is 20 ten thousand yuan, how many pieces of this product should the company produce to achieve the maximum profit, and what is the maximum profit?

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