
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 certain middle school went to a store to buy soccer balls and volleyballs. They bought $40$ soccer balls and $30$ volleyballs for a total of $4000$ yuan. It is known that buying one soccer ball costs $30$ yuan more than buying one volleyball.<br/>$(1)$ Find out how much it costs to buy one soccer ball and one volleyball each?<br/>$(2)$ The school decided to buy a total of $50$ soccer balls and volleyballs for the second time. It coincided with the store adjusting the prices of the items. The price of a soccer ball increased by $10\%$ compared to the first purchase, and a volleyball was sold at a discount of $10\%$ off the price of the first purchase. If the total cost of the second purchase of soccer balls and volleyballs is $86\%$ of the total cost of the first purchase, find out how many volleyballs the school bought for the second time.

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