
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 school plans to choose one of two sports equipment stores, Store A and Store B, to purchase a batch of table tennis rackets to enrich students' campus life. It is known that the price of each pair of table tennis rackets at Store A and Store B is $30. Now, the two stores have the following discount schemes: Store A sells at a 30% discount starting from the 11th pair if you buy 10 pairs or more; Store B sells at an 85% price starting from the 1st pair. Let the quantity of table tennis rackets the school plans to purchase be $x$ pairs $(x > 10$, and is a positive integer). The cost of purchasing from Store A is $y_{1}$ dollars, and the cost of purchasing from Store B is $y_{2}$ dollars.<br/>$(1)$ Please write down the relationship between $y_{1}$, $y_{2}$, and $x$ respectively;<br/>$(2)$ If the school plans to spend $1581$ dollars to purchase table tennis rackets, which store should the school choose to buy more table tennis rackets from?<br/>$(3)$ When $x > 12$, which sports equipment store is more cost-effective for the school to purchase from? Please explain the reason.

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