
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 with the following offers:<br/>Offer 1: Buy one tea set and get one tea bowl for free;<br/>Offer 2: Pay 95% of the original price for both the tea set and tea bowl.<br/>Now, a customer wants to buy 30 sets of tea sets and $x$ tea bowls ($x > 30$) from the mall.<br/>$(1)$ Express the amount to be paid for offers 1 and 2 with an algebraic expression involving $x$;<br/>$(2)$ If $x=40$, choose one of the two offers mentioned above, which one saves more money? Please calculate and explain;<br/>$(3)$ When $x=40$, can a more cost-effective option be found? If yes, provide your solution and calculate the amount to be paid for this option; if not, please explain why.

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