
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 Hunan, promoting health, and moving the spirit of Xiaoxiang," in preparation for the national online sports event in 2023, a certain middle school plans to purchase a batch of basketballs and volleyballs. If they buy $3$ basketballs and $1$ volleyball, the total cost is $360$ yuan; if they buy $5$ basketballs and $3$ volleyballs, the total cost is $680$ yuan. <br/>$(1)$ Find the price of each basketball and each volleyball in yuan. <br/>$(2)$ The school plans to purchase a total of $100$ basketballs and volleyballs, with the number of basketballs purchased being at least three times the number of volleyballs. How should they make the purchase to minimize the total cost? Also, find the minimum total cost.

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