
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 certain shopping mall originally sold a certain product with a cost price of $80$ yuan for $100$ yuan each, and could sell 100 units in a day. Later, after a market survey, it was found that for every $1$ yuan decrease in the unit price of the product, the sales volume would increase by 10 units.

(1) Find the profit the shopping mall could make in a day when operating the product at the original price.

(2) Let the product decrease in price by $x$ yuan later, and the shopping mall could make a profit of $y$ yuan in a day.

1. If the shopping mall wants to make a profit of $2160$ yuan in a day by operating the product, how much should the price of each product be decreased by?

2. Find the functional relationship between $y$ and $x$. For what value of $x$ will the shopping mall make the maximum profit?

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