
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 purchase some table tennis rackets and table tennis balls. A certain store sells a type of table tennis racket and table tennis ball, with each pair of table tennis rackets priced at $80$ yuan and each box of table tennis balls priced at $20$ yuan. During the National Day holiday, the store decides to launch a promotion offering two discount options to customers, namely:  
Option 1: Buy one pair of table tennis rackets and get one box of table tennis balls for free;  
Option 2: Pay $90\%$ of the original price for both the table tennis racket and the table tennis ball. The school wants to purchase 20 pairs of table tennis rackets and $x$ boxes of table tennis balls $(x > 20$, $x$ is an integer).  
$(1)$ If the school chooses Option 1, the payment will be ______ yuan; if the school chooses Option 2, the payment will be ______ yuan (expressed as an algebraic expression involving $x$);  
$(2)$ If $x=30$, please help calculate which option is more cost-effective at this time;  
$(3)$ If $x=30$, can a more cost-effective purchasing option be found? If yes, please write down the purchasing option and calculate the amount to be paid for this option; if not, please explain the reason.

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