
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 enrich campus sports life, the school has set up a tennis interest group and needs to purchase $30$ units of a certain brand of tennis training racket and $x$ cans of tennis balls $\left(x \gt 30\right)$. According to market research, the price of this brand of tennis racket is $100$ yuan per unit and tennis balls are $20$ yuan per can. There are two sports goods stores, Store A and Store B, with the following discount schemes:<br/>Store A: Buy one tennis racket and get one can of tennis balls for free;<br/>Store B: Pay $90\%$ of the total amount for both the tennis racket and tennis balls.<br/>$(1)$ Scheme 1: Buying from Store A requires paying ______ yuan (expressed as an algebraic expression involving $x$);<br/>Scheme 2: Buying from Store B requires paying ______ yuan (expressed as an algebraic expression involving $x).<br/>$(2)$ If $x=100$, please calculate and determine which scheme is more cost-effective for the school to adopt.<br/>$(3)$ If $x=100$, can you provide a more cost-effective purchasing plan? Describe your purchasing method and calculate how much money is saved.

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