
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 shopping mall sells a type of table tennis racket and table tennis balls. The table tennis racket is priced at $80$ yuan per pair, and the table tennis balls are priced at $20$ yuan per box. During the "National Day" holiday, the mall decided to launch a promotion, offering two types of discounts to customers.<br/>Option 1: Buy one table tennis racket and get one box of table tennis balls for free;<br/>Option 2: Pay $90\%$ of the original price for both the table tennis racket and the table tennis balls.<br/>A customer wants to buy 20 table tennis rackets and $x$ boxes of table tennis balls $(x > 20$ and is an integer).<br/>$(1)$ Express in algebraic terms the amount of money required to purchase under both options.<br/>$(2)$ If $x=30$, calculate and determine which option is more cost-effective;<br/>$(3)$ When $x=30$, can you come up with a more cost-effective purchasing plan? Please describe your purchasing method.

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