
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
A student conducted a market survey on the sales of a certain small commodity during the holiday, and found that the market price of the commodity on the same day is inversely proportional to his inventory quantity (in pieces) plus $20$. It is known that the purchase price of each item is $2. When he purchased $100$ pieces of this commodity, he sold out on the same day, making a profit of $100. If the commodities can be sold out every day, what is the maximum profit per day for this student? What is the inventory quantity when the maximum profit is achieved?

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