
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 certain supermarket in Beishan sells teapots and teacups. Each teapot is priced at $20$ yuan, and each teacup is priced at $6$ yuan. During the "Double Eleven" promotion period, the supermarket offers two discount options to customers:<br/>① Buy one teapot and get one teacup for free; ② Pay $90\%$ of the original price for both the teapot and teacup. Now, a customer wants to buy $5$ teapots and $x$ teacups (where $x$ is more than $5$).<br/>$(1)$ If the customer chooses option ① to purchase, the payment required is ______ yuan (expressed as an algebraic expression in terms of $x$); If the customer chooses option ② to purchase, the payment required is ______ yuan (expressed as an algebraic expression in terms of $x).<br/>$(2)$ If $x=20$, please calculate and explain which option is more cost-effective at this point?<br/>$(3)$ If $x=20$, considering both options ① and ②, can you design a more cost-effective purchasing strategy? Please write it out.

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