
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 department store sells a batch of brand-name shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to expand sales, increase profits, and reduce inventory quickly, the store decides to take appropriate price reduction measures. After investigation, it was found that if the price of each shirt is reduced by $1, the store can sell an average of 2 more shirts per day. Find:<br/>$(1)$ If the store needs to make an average daily profit of $1200, how much should the price of each shirt be reduced by?<br/>$(2)$ When the price of each shirt is reduced by how much, the store makes the maximum average daily profit?

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