
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
In the sales of a clothing store, it was found that a certain brand of children's clothing can sell an average of 40 pieces per day, with a profit of $80$ yuan per piece. In order to welcome the "National Day" holiday, the mall decided to take appropriate price reduction measures to increase sales, increase profits, and reduce inventory. According to market research, if the price of each piece of children's clothing is reduced by $4$ yuan, then an average of 8 more pieces can be sold per day.

$(1)$ To make a profit of $4800$ yuan per day on the sale of this type of children's clothing, how much should the price of each piece of clothing be reduced by?

$(2)$ If the company wants to maximize profit, how much should the price be reduced by? What is the maximum profit?

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