
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 store purchased a batch of stationery boxes for 1050 yuan, which sold out quickly. They then purchased a second batch for 1440 yuan, wherein the purchase price per box was 1.2 times that of the first batch, with 10 more boxes than the first batch.  
(1) What was the purchase price per box for the first batch?  
(2) After selling out the first batch, the second batch was sold at a price of 24 yuan per box. When exactly half of them were sold, the store decided to apply a one-time discount to sell the rest of the boxes due to market conditions, but required that the profit from this batch not be less than 288 yuan. What is the lowest discount that could be offered?

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