
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 store sells badminton rackets and shuttlecocks. The badminton rackets are priced at $40$ yuan per pair, and the shuttlecocks are priced at $10$ yuan per barrel. During the "Double Eleven" promotion, the store decided to offer two discount options to customers.<br/>Option 1: Buy one pair of badminton rackets and get one barrel of shuttlecocks for free;<br/>Option 2: Pay 90% of the original price for both the badminton rackets and shuttlecocks.<br/>Now, a customer wants to buy 10 pairs of badminton rackets and $x$ barrels of shuttlecocks $\left(x \gt 10\right)$.<br/>$(1)$ If the customer purchases according to Option 1 and Option 2, how much money will they need to pay? (Express your answers in terms of $x$)<br/>$(2)$ When $x=30$, by calculation, determine which option is more cost-effective at this point.<br/>$(3)$ When $x=30$, can you provide another purchasing option that saves more money? Describe your purchasing method and calculate the amount to be paid.

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