
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
In order to comprehensively promote quality education, enhance students' physical fitness, and enrich campus cultural life, our school will hold a special spring sports meeting. We need to purchase two types of prizes, $A$ and $B$. According to market research, if we buy 3 items of type $A$ and 2 items of type $B$, the total cost will be $60; if we buy 1 item of type $A$ and 3 items of type $B$, the total cost will be $55. <br/>$(1)$ Find the unit price of prizes $A$ and $B$ respectively; <br/>$(2)$ The organizing committee of the sports meeting plans to purchase a total of 100 items of prizes $A$ and $B$, with a total cost not exceeding $1160, and the quantity of type $A$ prizes is not more than 3 times the quantity of type $B$ prizes. How many purchasing plans does the organizing committee have? And find the minimum total cost.

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