
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 sportswear specialty store found that a certain type of sportswear has a cost price of $80$ yuan per piece and a selling price of $120$ yuan, and they can sell 20 pieces per day. In order to prepare for the National Day holiday, the store decided to take appropriate price reduction measures to increase sales volume and profits. Market research revealed that if the price of each piece of sportswear is reduced by $1$ yuan, then an average of 2 more pieces can be sold per day.  
$(1)$ Let the price reduction for each piece of sportswear be $x$ yuan, then the store can profit $120-x-80=(40-x)$ yuan per piece, and can sell $(20+2x)$ pieces per day. Express the answers in terms of $x$.  
$(2)$ If the specialty store plans to make an average profit of $1200$ yuan per day, how much should the price of each piece of sportswear be reduced by?

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