
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 shopping mall sells a batch of branded shirts, with an average daily sales volume of $20$ shirts, and a profit of $40$ yuan per shirt. In order to expand sales and increase profits, the mall decides to implement an appropriate price reduction strategy. After investigation, it was found that for every $1$ yuan reduction in price per shirt, the mall can sell an additional $2$ shirts on average.

$(1)$ If the price reduction per shirt is set at $x$ yuan, and the average daily profit is $y$ yuan, find the functional relationship between $y$ and $x$.
$(2)$ At what price reduction per shirt will the mall have the maximum average daily profit?
$(3)$ If the mall needs an average daily profit of $1200$ yuan, how much should the price per shirt be reduced?

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