
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
Yonghui supermarket sells teapots and teacups. Each teapot is priced at $20, and each teacup is $4. During this year's "Double Eleven" promotion, the supermarket will offer two discount options to customers:<br/>Option 1: Buy one teapot and get one teacup for free;<br/>Option 2: Pay 90% of the regular price for both the teapot and teacup.<br/>A customer plans to buy 5 teapots and $x$ teacups (where $x$ is more than 5) at the supermarket.<br/>$(1)$ Express in algebraic terms containing $x$ how much money is required for each of the two options?<br/>$(2)$ When $x=20$, calculate and explain which of the two purchasing options the customer should choose to save more money.<br/>$(3)$ When $x=20$, can you provide a more cost-effective purchasing plan? Please write down your purchasing plan.

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