
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 appliance store sells a type of washing machine and an induction cooker. The washing machine is priced at $800$ yuan each, and the induction cooker is priced at $200$ yuan each. During the "National Day" holiday, the store decided to launch a promotion, offering two discount options to customers:<br/>Option 1: Buy one washing machine and get one induction cooker for free;<br/>Option 2: Both the washing machine and the induction cooker are paid at $90\%$ of the original price.<br/>Now, a customer wants to buy $10$ washing machines and $x$ induction cookers from the store $\left(x \gt 10\right)$.<br/>$(1)$ If the customer purchases according to Option 1 and Option 2, how much money does the customer need to pay in each case? (Express in terms of $x$)<br/>$(2)$ Find the value of $x$ when the costs of Option 1 and Option 2 are the same.<br/>$(3)$ When $x=40$, can you come up with a more cost-effective purchasing plan? Describe your purchasing method and calculate how much money needs to be paid.

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