
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 shopping mall sells a type of amplifier and a page-turning pen. The amplifier is priced at $300$ yuan each, and the page-turning pen is priced at $50$ yuan each. During the National Day holiday, the mall launched a promotion offering two discount options simultaneously. Option 1: Buy one amplifier and get one page-turning pen for free. Option 2: Both the amplifier and the page-turning pen are priced at $90\%$ of their regular price. Now, a customer goes to the mall to buy 20 amplifiers and $x$ page-turning pens $(x$ exceeds $20)$. <br/> $(1)$ If the customer chooses option 1, find the amount the customer needs to pay (expressed in terms of $x$); <br/> $(2)$ If the customer chooses option 2, find the amount the customer needs to pay (expressed in terms of $x$); <br/> $(3)$ When $x=30$, can you come up with a more cost-effective purchasing option? Describe your purchasing method and calculate the amount required.

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