
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
Jiale Supermarket sells teapots and teacups. Each teapot is priced at $90, and each teacup is priced at $25. During the "Double Eleven" promotion, the supermarket offers two discount options to customers:<br/>Option 1: Buy one teapot and get one teacup for free;<br/>Option 2: Pay 90% of the original price for both the teapot and teacup.<br/>During the promotion, a customer wants to buy 5 teapots and $x$ teacups $\left(x \gt 5\right)$.<br/>$(1)$ Calculate the amount of money the customer needs to pay for each option (expressed as algebraic expressions with $x$).<br/>$(2)$ When $x=30$, calculate and determine which option is more cost-effective.<br/>$(3)$ When $x=30$, can you design a more cost-effective purchasing method by combining these two discount options? Please write it out.

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