
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
To further implement the spirit of exercising for at least 1 hour every day, a certain middle school plans to purchase a certain number of soccer balls and basketballs from a mall in one go, for use in school ball games. The price of each soccer ball and basketball is the same. It is known that the price of a basketball is $30 less than twice the price of a soccer ball. It is also known that the quantity of soccer balls purchased with $1200 is twice the quantity of basketballs purchased with $900.

$(1)$ What are the prices of a soccer ball and a basketball, respectively?

$(2)$ Based on the actual situation of the school, a total of 100 soccer balls and basketballs need to be purchased at once. It is required that the quantity of soccer balls purchased is no more than half the quantity of basketballs purchased, and the total cost of purchasing soccer balls and basketballs is less than $8100. What are the possible purchasing plans for the school?

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