
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
Meixian, Shaanxi is the largest high-quality kiwi production base in China, known as the "hometown of Chinese kiwi." A merchant purchased a batch of kiwis from Meixian for sale (only sold by the whole box, not retail). The purchase price is $80$ yuan per box. When the selling price is $120$ yuan per box, 20 boxes can be sold per day. A market survey found that if the price of each box of kiwi is reduced by $1$, then an average of 2 more boxes can be sold per day.

$(1)$ By how much should the price of each box of kiwi be reduced for the merchant to make a daily profit of $1200$ yuan?

$(2)$ Can the merchant's average daily profit reach $1800$ yuan? Please explain your reasoning.

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