
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]


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In order to cultivate students with comprehensive development in morality, intelligence, physical fitness, aesthetics, and labor, a certain school is preparing to purchase $50$ soccer balls and $x$ solid balls $\left(x \gt 50\right)$ to enhance students' physical fitness. The price of a soccer ball is $80$ yuan per ball, and the price of a solid ball is $20$ yuan per ball. Two stores, Store A and Store B, have provided their respective discount plans to the school:<br/>Store A: Buy one soccer ball and get one solid ball for free;<br/>Store B: Pay $90\%$ of the price for both the soccer ball and the solid ball.<br/>$(1)$ If the school purchases from Store A and Store B respectively, how much money will they need to pay? (Express in algebraic expression with $x$)<br/>$(2)$ If $x=200$, calculate and determine which store is more cost-effective at this point.<br/>$(3)$ When $x=300$, can you provide a more cost-effective purchasing plan? Describe your purchasing method and calculate the payment amount.

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