
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 department store sells suits and ties, with each suit priced at $400$ yuan and each tie priced at $80 yuan. During the National Day holiday, the store decided to launch a promotion offering two discount options to customers:<br/>Option 1: Buy one suit and get one tie for free;<br/>Option 2: Both the suit and tie are priced at $90\%$ of the original price.<br/>Now a customer wants to buy $20$ suits and $x$ ties $\left(x \gt 20\right)$.<br/>$(1)$ If the customer chooses Option 1, how much money will they need to pay (expressed in terms of $x$)? If the customer chooses Option 2, how much money will they need to pay (expressed in terms of $x$)?<br/>$(2)$ If $x=30$, calculate and determine which option is more cost-effective at this point.<br/>$(3)$ When $x=30$, can you provide a more cost-effective purchasing method? Please write down your purchasing method and calculate the required cost.

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