
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 certain shop is preparing to purchase a batch of meat zongzi and honey date zongzi before the Dragon Boat Festival. It is known that the unit price of meat zongzi is $2.5$ yuan more than the unit price of honey date zongzi, and the number of meat zongzi that can be bought for $300$ yuan is exactly twice the number of honey date zongzi that can be bought for $100$ yuan.  
$(1)$ Find the unit prices of meat zongzi and honey date zongzi.  
$(2)$ If the shop purchases 100 honey date zongzi and 200 meat zongzi at once, and sells them at prices of $6$ yuan per piece and $10$ yuan per piece respectively in the following manner: the price of meat zongzi increases by $10\%$ before the Dragon Boat Festival, and after the festival, the price of meat zongzi is discounted by $10\%$, while the price of honey date zongzi remains unchanged. If a total profit of $570$ yuan is made after selling all the zongzi, find the number of meat zongzi sold before the Dragon Boat Festival.

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