
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 inaugural Sea Holly Cultural Festival and the on-site release ceremony of the original campus song "No. 86 Jingguang Road" of the school were held at the Jingguang campus on April 2nd. The Sea Holly Cultural Festival series of activities also started simultaneously with the launch ceremony. To reward classes and individuals who actively participate in the activities, the school plans to purchase two types of prizes, $A$ and $B$. If purchasing 3 of prize $A$ and 2 of prize $B$ costs a total of $130, and if purchasing 5 of prize $A$ and 4 of prize $B$ costs a total of $230. Find:  
$(1)$ The unit price of prizes $A$ and $B$.  
$(2)$ According to the school's plan, prepare to purchase a total of 20 prizes of types $A$ and $B$, with the quantity of prize $A$ being at least twice the quantity of prize $B$. Design the most cost-effective purchasing plan and explain the reasoning.

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