
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 large shopping mall sells a set of tea sets and tea bowls. The tea set is priced at $200 per set, and the tea bowl is priced at $20 each. During the "Double Eleven" period, the mall decided to launch a promotion. During the promotion, two discount options were offered to customers: Option 1: Buy a tea set and get a tea bowl for free; Option 2: Pay 95% of the total price for the tea set and tea bowl. Now, a customer wants to buy 30 sets of tea sets and $x$ tea bowls ($x > 30$).

$(1)$ If the customer chooses Option 1, the payment required is ______ yuan; if the customer chooses Option 2, the payment required is ______ yuan. (Express in algebraic expressions involving $x$)<br/>$(2)$ If $x=40$, explain through calculations which purchasing option is more appropriate at this time.<br/>$(3)$ When $x=40$, can a more cost-effective option be found? If yes, state your option and calculate the amount to be paid for this option; if not, explain the reason.

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