
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. Each suit is priced at $1000$ yuan, and each tie is priced at $200 yuan. During the "Double Eleven" promotion, the store decides to offer two discount options to customers.<br/>Option 1: Buy one suit and get one tie for free;<br/>Option 2: All items are sold at a 10% discount. Now, a customer wants to buy 10 suits and $x$ ties $\left(x \gt 10\right)$.<br/>$(1)$ If the customer chooses Option 1, the payment will be ______ yuan (expressed as an algebraic expression in terms of $x$); if the customer chooses Option 2, the payment will be ______ yuan (expressed as an algebraic expression in terms of $x$);<br/>$(2)$ If $x=30$, by calculation, determine which purchasing method, Option 1 or Option 2, is more cost-effective;<br/>$(3)$ When $x=30$, can you come up with a more cost-effective purchasing plan? Write down your purchasing method and calculate the total payment.

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