
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
The 19th Asian Games will be held in Hangzhou, China on September 23, 2023. A certain store found that when each set of Asian Games mascot badges earns a profit of $40, they can sell 20 sets per day. In order to welcome the Asian Games, the store decided to take appropriate price reduction measures. After investigation, it was found that if the selling price is reduced by $1, the store will sell 2 more sets on average per day.

$(1)$ When each set of badges earns a profit of $38, how many more sets can be sold per day?

$(2)$ In order to reduce inventory as soon as possible, how much should the price of each set of mascot badges be reduced so that the daily profit from selling mascot badges in the store reaches $1200?

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