
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 sports equipment store sells a certain brand of tennis rackets and tennis balls. The tennis rackets are priced at $200$ yuan per pair, and the tennis balls are priced at $40$ yuan per box. During the National Day holiday, the sports equipment store decided to launch a promotion, offering two discount options to consumers:<br/>Option 1: Buy one pair of tennis rackets and get one box of tennis balls for free.<br/>Option 2: Pay $90\%$ of the original price for both the tennis rackets and tennis balls.<br/>Now, a school wants to purchase $20$ pairs of tennis rackets and $x$ boxes of tennis balls, where $x \gt 20$.<br/>$(1)$ If the school purchases according to Option 1, the payment will be ______ yuan; if they purchase according to Option 2, the payment will be ______ yuan. (Express the results in terms of $x$ and simplify the expressions.)<br/>$(2)$ If $x=30$, calculate and determine which option is more cost-effective at this point.<br/>$(3)$ When $x=30$, can you come up with a more cost-effective purchasing plan? Describe your purchasing method and the required cost.

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