
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
The Red Scarf Gym plans to purchase a certain brand of table tennis rackets and table tennis balls. It is known that the table tennis rackets of this brand are priced at $150$ yuan per pair, and the table tennis balls are priced at $15$ yuan per box. During the New Year period, this brand has decided to launch a promotional campaign, offering two discount options to customers, namely:  
Option one: Buy one pair of table tennis rackets and get two boxes of table tennis balls for free;  
Option two: Pay 90% of the total price for both the table tennis rackets and table tennis balls at the regular price.  
The gym plans to purchase 10 pairs of table tennis rackets and $x$ boxes of table tennis balls $(x > 20$, $x$ is an integer).  
$(1)$ When $x=40$, if the gym purchases according to option one, the payment will be ______ yuan; if the gym purchases according to option two, the payment will be ______ yuan;  
$(2)$ For what value of $x$ will the costs be the same under both purchasing options?  
$(3)$ If $x=40$, can a more cost-effective purchasing option be found? If yes, please write down the purchasing option and calculate the cost; if not, please explain the reason.

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