
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 product has a cost price of $40$ yuan per unit. When the selling price is $60$ yuan per unit, 300 units can be sold per week. It is now necessary to reduce the price for clearance. According to market research, for every $1$ yuan reduction in price, an additional 20 units can be sold per week. Answer the following questions under the premise of ensuring profitability:

1. If the price reduction per unit is $x$ yuan and the profit from selling the goods per week is $y$ yuan, write the function relationship between $y$ and $x$, and determine the range of values for the independent variable $x$.
2. How much should the price be reduced by to maximize the profit per week? What is the maximum profit?

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