
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
With the implementation of the "double reduction" policy, middle school students have more free time for outdoor sports. For this reason, a certain school decides to purchase a batch of sports equipment. It is known that the unit price of a soccer ball is $30$ yuan more than the unit price of a volleyball. The school purchased a volleyball for $500$ yuan, and the quantity of volleyball is the same as the quantity of soccer balls purchased for $800$ yuan.
$(1)$ What are the unit prices of the volleyball and soccer ball?
$(2)$ If the school plans to purchase a total of $11$ volleyball and soccer balls, with no fewer than $2$ soccer balls. Let the cost of purchasing volleyball and soccer balls be $y$ yuan, and let the quantity of volleyball be $x$. Find the functional relationship between $y$ and $x$, and design a purchasing plan with the minimum cost. Write down the minimum cost.

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