
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 shopping mall purchased a batch of air conditioners for $24000$ yuan and then sold them at a price of $3000$ yuan per unit. Due to the hot weather, the air conditioners were quickly sold out. The shopping mall then purchased the same model of air conditioners again at a price of $52000$ yuan. The quantity was twice that of the first purchase, but the unit price was increased by $200$ yuan, and the selling price was also increased by $200$ yuan.
$(1)$ What was the unit cost of the air conditioners during the first purchase?
$(2)$ In order to sell out the second batch of air conditioners as soon as possible and maintain a profit rate of no less than $22\%$ for the sales of both batches, the shopping mall decided to sell part of the second batch at a $95\%$ discount. What is the maximum number of air conditioners that can be sold at a discount?

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