
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
The annual $NK$ school celebration and sports day have successfully concluded. To commend students who have performed outstandingly during the celebration and sports day, the class committee of a certain 8th-grade class purchased two types of cultural and creative products, $A$ and $B$, as prizes in two separate instances. The unit price for each type of prize was the same for both purchases. The specific details are shown in the table below:

|           | Quantity Purchased (pieces) | Total Cost (yuan) |
|-----------|-----------------------------|-------------------|
|           | $A$                         | $B$               |
| First Time| $15$                        | $20$              | $520$             |
| Second Time| $20$                       | $17$              | $616$             |

$(1)$ Find the unit prices of prizes $A$ and $B$ in yuan.  
$(2)$ Considering that the cheerleading team and logistics service students have also made significant contributions, the class committee plans to purchase another batch of prizes for no more than $480$ yuan. They aim to purchase a total of $100$ pieces of prizes $A$ and $B$ in three rounds, with the quantity of prizes $A$ purchased in the third round being no less than the quantity of prizes $B$. What are the possible purchasing schemes for the third round?

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