
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
A school plans to purchase two brands of soccer balls, brand A and brand B. It is known that the unit price of brand A soccer balls is $30 less than the unit price of brand B soccer balls. The quantity of brand A soccer balls that can be purchased with $1000 is the same as the quantity of brand B soccer balls that can be purchased with $1600.<br/>$(1)$ Find the unit prices of brand A and brand B soccer balls.<br/>$(2)$ The school plans to purchase a total of 80 soccer balls of both brand A and brand B. Let $a$ be the number of brand A soccer balls to be purchased, where the quantity of brand A soccer balls is at least 30 but not more than 3 times the quantity of brand B soccer balls. Determine how many brand A soccer balls should be purchased to minimize the total cost $W$.

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