
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
Xibing is a local specialty in Haiyang, with a unique flavor, symbolizing joy and reunion. Person A and person B went to the market to purchase the same kind of gift box filled with Xibing at the same price. Person A bought $2400$ yuan worth of Xibing, which was $10$ boxes less than what person B bought for $3000$ yuan.<br/>$(1)$ Using fractional equations, find the quantity of Xibing that person A purchased;<br/>$(2)$ When person A and person B went to purchase the same kind of gift box filled with Xibing again, they coincidentally encountered a store promotion where the unit price was $20$ yuan less per box compared to the previous purchase. Person A spent the same total amount on Xibing as before, while person B bought the same quantity as before. Then, the average unit price of Xibing for person A over the two purchases is ______ yuan per box, and for person B is ______ yuan per box (write down the answers directly).

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