
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 school plans to purchase prizes for the "Me and My Motherland" speech competition. It is known that purchasing $3$ type $A$ prizes and $2$ type $B$ prizes costs a total of $120$ yuan; purchasing $5$ type $A$ prizes and $4$ type $B$ prizes costs a total of $210$ yuan.  
$(1)$ Find the unit prices of type $A$ and type $B$ prizes.  
$(2)$ The school plans to purchase a total of $30$ type $A$ and type $B$ prizes, with the quantity of type $A$ prizes not less than $\frac{1}{3}$ of the quantity of type $B$ prizes. Design the most cost-effective purchasing plan and explain the reasoning.

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